Math 232 Calculus 2 - Spring 2016

§6.1 - AREA BETWEEN CURVES

§6.1 - Area Between Curves

Recall: to compute the area below a curve y = f (x), between x = a and x = b, we candivide up the region into rectangles.

The area of one small rectangle is

The approximate area under the curve is

The exact area under the curve is

2


To compute the area between the curves y = f (x) and y = g(x), between x = a andx = b, we can divide up the region into rectangles.


The approximate area between the two curves is

The exact area between the two curves is

This formula works as long as f (x) g(x).

3


Example. Find the area between the curves y = x2 + x and y = 3 − x2

4


Review. The area between two curves y = f (x) and y = g(x) between x = a and x = bis given by:

5


Review. The area between the curves y = 2x + 1 and y = 5 − 2x2 is given by:

A.∫ 1−2

2x + 1 − 5 + 2x2 dx

B.∫ 1−2

5 − 2x2 − 2x + 1 dx

C.∫ 1−2

5 − 2x2 − 2x − 1 dx

D.∫ 5−3

5 − 2x2 + 2x + 1 dx

E. None of these.

6


Example. The area between the curves y = cos(5x), y = sin(5x), x = 0, and x = π4 isgiven by:

A.∫ π/4

0sin(5x) − cos(5x) dx

B.∫ π/4

0cos(5x) − sin(5x) dx

C. Both of these answers are correct.

D. Neither of these answers are correct.

7


Example. Set up the integral to find the area bounded by the three curves in the centerof the figure shown.

• f (x) = x2 − x − 6• g(x) = x − 3• h(x) = −x2 + 4

8


Note. The area between two curves x = f (y) and x = g(y) between y = c and y = d isgiven by:

This formula works as long as f (y) g(y).

9


To compute the area between the curves x = f (y) and x = g(y), between y = c andy = d, we can again divide up the region into rectangles.


The approximate area between the two curves is

The exact area between the two curves

10


Example. Find the area between the curves f (y) = sin(y)+5, g(y) =y2

√36 + y3

6, y = −2,

and y = 2.

11


Example. The area between the curves y = x2 and y = 3x2, and y = 4 is given by:

A.∫ 2

0x2 − 3x2 dy

B.∫ 4

03x2 − x2 dy

C.∫ 2

0

√y −

√y3

dy

D.∫ 4

0

√y −

√y3

dy

E.∫ 4

0

√y3− √y dy

12


Extra Example. Find the area between the curves 2x = y2 − 4 and y = −3x + 2 that liesabove the line y = −1

13


Extra Example. In the year 2000, the US income distribution was: (data from WorldBank, see http://wdi.worldbank.org/table/2.9)

Income Category Fraction of Fraction of Cumulative CumulativePopulation Total Income Fraction of Fraction of

Population IncomeBottom 20% 0.20 0.05 0.20 0.05

2nd 20% 0.20 0.11 0.40 0.163th 20% 0.20 0.16 0.60 0.324th 20% 0.20 0.22 0.80 0.54

Next 10% 0.10 0.16 0.90 0.70Highest 10% 10 0.30 1.00 1.00

The Lorenz curve plots the cumulative fraction of population on the x-axis and thecumulative fraction of income received on the y-axis.

The Gini index is the area between the Lorenz curve and the line y = x, multiplied by2.Estimate the Gini index for the US in the year 2000 using the midpoint rule.

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http://wdi.worldbank.org/table/2.9

§6.2 - VOLUMES

§6.2 - Volumes

If you can break up a solid into n slabs, S1,S2, . . .Sn, each with thickness ∆x, then

Volume of solid ≈

The thinner the slices, the better the approximation, so

Volume of solid =

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§6.2 - VOLUMES

Example. Find the volume of the solid whose base is the ellipsex2

4+

y2

9= 1 and whose

cross sections perpendicular to the x-axis are squares.

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§6.2 - VOLUMES

Volumes found by rotating a region around a line are called solids of revolution.

For solids of revolution, the cross sections have the shape of a or theshape of a .

The area of the cross-sections can be described with the formulas

The volume of a solid of revolution can be described with the formulas:

When the region is rotated around the x-axis, or any other horizontal line, then weintegrate with respect to .When the region is rotated around the y-axis, or any other vertical line, then weintegrate with respect to .

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§6.2 - VOLUMES

Example. Consider the region bounded by the curve y = 3√

x, the x-axis, and the linex = 8. What is the volume of the solid of revolution formed by rotating this regionaround the x-axis?

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§6.2 - VOLUMES

Example. Consider the region in the first quadrant bounded by the curves y = 3√

x andy = 14x. What is the volume of the solid of revolution formed by rotating this regionaround the x-axis? The y-axis?

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§6.2 - VOLUMES

Review. Suppose a 3-dimensional solid can be sliced perpendicular to the x-axis andthe slice at position x has area given by the function A(x). Then the volume is givenby:

Review. If the volume is a solid of revolution, then the volume is given by:

Question. Which of the following is NOT a solid of revolution?A. a bowl of soup B. a watermelon C. a square cake D. a bagel

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§6.2 - VOLUMES

Example. The region between the curves y = ex, x = 0, and y = e3 is rotated aroundthe x-axis, to make a solid of revolution. When computing the volume, what are thecross-sections and which variable do we integrate with respect to?

A. cross-sections are disks, integrate with respect to dx

B. cross-sections are disks, integrate with respect to dy

C. cross-sections are washers, integrate with respect to dx

D. cross-sections are washers, integrate with respect to dy

Set up an integral to calculate the volume.

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§6.2 - VOLUMES

Example. The region between the curve y = ex, x = 0, and y = e3 is rotated aroundthe y-axis, to make a solid of revolution. When computing the volume, what are thecross-sections and which variable do we integrate with respect to?

A. cross-sections are disks, integrate with respect to dx

B. cross-sections are disks, integrate with respect to dy

C. cross-sections are washers, integrate with respect to dx

D. cross-sections are washers, integrate with respect to dy

Set up an integral to calculate the volume.

Set up an integral to calculate the volume if this region is rotated around the line x = −5instead of the y-axis.

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§6.2 - VOLUMES

Extra Example. Consider the region bounded by y = 6x2 , x = 1, x = 2, and the x-axis.

Set up an integral to compute the volume of the solid obtained by rotating this regionabout the line x = 1.

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§6.2 - VOLUMES

Example. Find the volume of the solid whose base is the region between y =√

x, thex-axis, and the lines x = 1 and x = 5, and whose cross sections perpendicular to thex-axis are equilateral triangles.

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§6.2 - VOLUMES

Example. Find the volume of the solid whose base is the region between y =√

x, thex-axis, and the lines x = 1 and x = 5, and whose cross sections perpendicular to they-axis are equilateral triangles.

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§6.2 - VOLUMES

Example. Find the volume of a pyramid with a square base of side length b and heighth.

26

§6.2 - VOLUMES

Example. Find the volume of a cone with a circular base of radius a and height h.

27

§6.2 - VOLUMES

Extra Example. Set up an integral to find the volume of a bagel, given the dimensionsbelow.

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§6.4 - WORK

§6.4 - Work

Definition. If if a constant force F is applied to move an object a distance d, then thework done to move the object is defined to be

Question. What are the units of force? What are the units of work?

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§6.4 - WORK

Note. Suppose a Calculus book is 2 pounds (US units), which is 0.9 kg (metric units).The pounds is a unit of . The kg is a unit of .

The force on the book is in US units, or in metric units.

Example. How much work is done to lift a 2 lb book off the floor onto a shelf that is 5feet high?

Example. How much work is done to lift a 0.9 kg book off the floor onto a shelf that is1.5 meters high?

30

§6.4 - WORK

Example. A particle moves along the x axis from x = a to x = b, according to a forcef (x). How much work is done in moving the particle? (Note: the force is not constant!)

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§6.4 - WORK

Example. How much work is required to lift a 1000-kg satellite from the earth’s surfaceto an altitude of 2 · 106 m above the earth’s surface?The gravitational force is F =

GMmr2

, M is the mass of the earth, m is the mass of thesatellite, and r is the distance between the satellite and the center of the earth, and G isthe gravitational constant.

The radius of the earth is 6.4 ·106 m, its mass is 6 ·1024 kg, and the gravitational constant,G, is 6.67 · 10−11.Reference https://www.physicsforums.com/threads/satellite-and-earth.157112/

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§6.4 - WORK

Review. In the expression W =∫ b

aF(x) dx what do W, F(x), and dx represent?

Review. Which of the following statements are true:

A) If you are told that an object is 5 kg, and you want the force due to gravity (inmetric units), you need to multiply by g = 9.8m/s2.

B) If you are told that an object is 5 lb, and you want the force due to gravity (inEnglish units), you need to multiply by 32 f t/s2.

C) Both.

D) Neither.

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§6.4 - WORK

Example. In the alternative universe of the Golden Compass, the souls of humans andtheir animal companions, called daemons, are closely tied. Suppose that the forceneeded to separate a human and its daemon is given by f (x) = 10xe−x

2/1000 pounds,where x represents the distance between the human and the daemon in feet. Lyra andher daemon are currently 5 feet apart. How much work will it take to separate theman additional 5 feet?

34

§6.4 - WORK

Example. (Example 4 in book) A 200-kg cable is 300 m long and hangs vertically fromthe top of a tall building. How much work is required to lift the cable to the top of thebuilding?

What if we just needed to lift half the cable?

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§6.4 - WORK

Example. An aquarium has a square base of side length 4 meters and a height of 3meters. The tank is filled to a depth of 2 m How much work will it take to pump thewater out of the top of the tank through a pipe that rises 0.5 meters above the top ofthe tank?

36

§6.4 - WORK

Extra Example. A bowl is shaped like a hemisphere with radius 2 feet, and is full ofwater. How much work will it take to pump the water out of the top of the bowl? Usethe fact that water weights 62.5 pounds per cubic foot.

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§6.4 - WORK

Springs follow Hooke’s law: the force required to stretch them a distance x past theirequilibrium position is given by f (x) = kx, where k is a constant that depends on thespring.

Example. A spring with natural length 15 cm exerts a force of 45 N when stretched toa length of 20 cm.

1. Find the spring constant

2. Set up the integral/s needed to find the work done in stretching the spring 3 cmbeyond its natural length.

3. Set up the integral/s needed to find the work done in stretching the spring from alength of 20 cm to a length of 25 cm.

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§6.5 - AVERAGE VALUE OF A FUNCTION

§6.5 - Average Value of a Function

To find the average of a list of numbers q1, q2, q3, . . . , qn, we sum the numbers and divideby n:

For a continuous function f (x) on an interval [a, b], we could estimate the averagevalue of the function by sampling it at a bunch of evenly spaced x-values c1, c2, . . . , cn,which are spaced ∆x apart:

average ≈

The approximation gets better as n→∞, so we can define

average = limn→∞

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The resulting formula is analogous to the formula for an average of a list of numbers,since taking an integral is analogous to , and dividing by the length of theinterval b − a is analogous to dividing by .Example. Find the average value of the function g(x) =

11 − 5x on the interval [2, 5].

Is there a number c in the interval [2, 5] for which g(c) equals its average value? If so,find all such numbers c. If not, explain why not.

40


Question. Does a function always achieve its average value on an interval?

Theorem. (Mean Value Theorem for Integrals) For a continuous function f (x) on an interval

[a, b], there is a number c with a ≤ c ≤ b such that f (c) =∫ b

a f (x)dx

b − a .

Proof:

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Review. The average value of a function f (x) on the interval [a, b] is defined as:

and the Mean Value Theorem for Integrals says that:

42


Example. For the function f (x) = sin(x),

a) Find its average value on the interval [0, π].

b) Find any values c for which f (c) = fave. Give your answer(s) in decimal form.

c) Graph the curve f (x) = sin(x) and on your graph draw a rectangle of area equal tothe area under the curve from 0 to π.

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Example. Suppose g(x) is a continuous function and∫ 5

2g(x) dx = 12. Which of the

following are necessarily true?

A. For some number x between 2 and 5, g(x) = 4.

B. For some number x between 2 and 5, g(x) = 3.

C. For some number x between 2 and 5, g(x) = 5.

D. All of these are necessarily true.

E. None of these are necessarily true.

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Extra Example. The temperature on a July day starts at 60◦ at 8 A.M., and rises (withoutever falling) to 96◦ at 8 P.M.

1. Why can’t you say with certainty that the average temperature between 8 A.M.and 8 P.M. was 78◦?

2. What can you say about the average temperature during this 12-hour period?

3. Suppose you also know that the average temperature during this period was 84◦.Is it possible that the temperature was 80◦ at 6 P.M.? At 4 P.M.?

45

§7.1 - INTEGRATION BY PARTS

§7.1 - Integration by Parts

Recall: the Product Rule says:

Rearranging and integrating both sides gives the formula:

Note. This formula allows us to rewrite something that is difficult to integrate in termsof something that is hopefully easier to integrate. Integrating using this method iscalled:

46


Example. Find∫

xex dx.

47


Review. ∫u dv =

Example. (# 13) Integrate∫

t sec2(2t) dt using integration by parts. What is a goodchoice for u and what is a good choice for dv?

48


Example. Find∫

x(ln x)2dx

49


Example. Integrate∫ 2

1arctan(x)dx.

50


Example. Find∫

e2x cos(x)dx.

51


Question. How do we decide what to call u and what to call dv?

Question. Which of these integrals is a good candidate for integration by parts? (Morethan one answer is correct.)

A.∫

x3 dx

B.∫

ln(x) dx

C.∫

x2ex dx

D.∫

xex2 dx

E.∫

ln y√

ydy

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§7.2 - INTEGRATING TRIG FUNCTIONS

§7.2 - Integrating Trig Functions

Note. Here are some useful trig identities for the next few sections.

1. Pythagorean Identity:

2. Converted into tan and sec:

3. Converted into cot and csc:

4. Even and Odd:

5. Angle Sum Formula: sin(A + B) =

6. Angle Sum Formula: cos(A + B) =

7. Double Angle Formula: sin(2θ) =

8. Double Angle Formulas: cos(2θ) =

9.

10.

11. cos2(θ) =

12. sin2(θ) =

53


Example. Find∫

sin4(x) cos(x) dx

Example. Find∫

sin4(x) cos3(x) dx

54


Example. Find∫

sin5(x) cos2(x) dx

55


Example. Consider ∫cos2(x)dx

.

According to a TI-89 calculator∫cos2(x) dx =

sin(x) cos(x)2

+x2

.

According to the table in the back of the book,∫cos2(x) dx =

12

x +14

sin 2x

.

Are these answers the same?

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Compute∫

cos2(x) dx by hand. Hint: cos2(x) =1 + cos(2x)

2

Example. Compute∫

sin2(x) dx by hand.

57


Example. Compute∫

sin6(x) dx by hand.

58


Review. What tricks can be used to calculate∫

cos7(5x) sin4(5x) dx?

59


Which of these integrals can be attacked in the same way, using the identitysin2(x) + cos2(x) = 1 and u-substitution?

A.∫

sin3(x) cos4(x) dx

B.∫

sin4(x) cos5(x) dx

C.∫

sin3(x) cos5(x) dx

D.∫

cos2(x) sin4(x) dx

E.∫

cos7(8x) dx

F.∫

cos3(√

x)√x

dx

G.∫

sin3(2x)√

cos(2x) dx

H.∫

tan3(x) dx

I.∫

sin2(x) dx

60


Even powers of sine and cosine.

Review. What trig identities are most useful in evaluating∫

cos2(x) sin4(x) dx?

Example. Compute∫

cos2(x) sin4(x) dx by hand.

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Conclusions:

To find∫

sinm(x) cosn(x) dx,

if m is odd and n is even:

if n is odd and m is even:

if both m and n are odd:

if both m and n are even:

62


Note. Often the answers that you get when you integrate by hand do not look identicalto the answers you will see if you use your calculator, Wolfram Alpha, or the integraltable in the back of the book. Of course, the answers should be equivalent. Why doyou think the answers look so different?

63


These integrals have their own special tricks.

Example.∫

tan2(x) dx

Example.∫

sec(x) dx

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§7.3 - TRIG SUBSTITUTIONS

§7.3 - Trig Substitutions

The following three trig identities are useful for doing trig substitutions to solve somekinds of integrals with square roots in them.

sin2(x) + cos2(x) = 1 tan2(x) + 1 = sec2(x) cot2(x) + 1 = csc2(x)

65


Example. According to Wolfram Alpha,∫x2√

49 − x2dx =

12

(49 sin−1

(x7

)− x√

49 − x2)

Let’s see where that answer comes from using a trig substitution.

66


Review. To compute∫

x2√49−x2

dx, which trig identity is most useful?

A. sin2(θ) + cos2(θ) = 1

B. tan2(θ) + 1 = sec2(θ)

C. sin2(θ) = 12 − cos(2θ)D. sin(2θ) = 2 sin(θ) cos(θ)

67


Example. Find∫

1√x2 + a2

dx. (Assume a is positive.)

68


Example. Compute the integral∫ 2/3

1/3

√9x2 − 1

x2dx

69


Which trig substitutions for which problems?

70


What trig substitutions would be most useful for these integrals?

1.∫

2√4 + x2

dx

2.∫

(100x2 − 1)3/2 dx

3.∫

x

√4 − x

2

9dx

4.∫

(25 − x2)2 dx

5.∫ √

−x2 − 6x + 7 dx

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Extra Example. Use calculus to find the volume of a torus with dimensions R and r asshown.

72

§7.4 - INTEGRALS OF RATIONAL FUNCTIONS

§7.4 - Integrals of Rational Functions

Example. According to Wolfram Alpha,∫3x + 2

x2 + 2x − 3 dx =54

ln |1 − x| + 74

ln |x + 3| + C

Let’s see where this answer came from.

73


Review. True or False:∫

12x2 − 7x − 4 dx = ln |2x

2 − 7x − 4| + C

74


Example. (a) Find∫

12x2 − 7x − 4 dx (b) Find

∫3x − 5

2x2 − 7x − 4 dx

75


Example. Find∫

2x2 + 7x + 19x2 − 5x + 6 dx

76


Example. How would you set up partial fractions to integrate this?∫5x + 7

(x − 2)(x + 5)(x) dx

77


Example. How would you set up partial fractions to integrate this?∫

4x2 + 3x + 7x3 − 4x2 + 4xdx

A.4x2 + 3x + 7

x(x − 2)2 =Ax

+B

x − 2

B.4x2 + 3x + 7

x(x − 2)2 =Ax

+B

(x − 2)2

C.4x2 + 3x + 7

x(x − 2)2 =Ax

+B

x − 2 +C

(x − 2)2

D.4x2 + 3x + 7

x(x − 2)2 =Ax

+Bx + C(x − 2)2

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§7.5 INTEGRATION STRATEGIES

§7.5 Integration Strategies

For each integral, indicate what technique you might use to approach it and give thefirst step. You do not need to finish any of the problems.

1.∫

x3 ln x dx

2.∫

cos2(x) dx

3.∫

dxx ln(x)

4.∫

arcsin(x) dx

5.∫

x2 + 1√x

dx

6.∫

sin(x)3 + sin2(x)

dx

7.∫

x3

25 − x2 dx

8.∫

x3√25 − x2

dx

9.∫

x + 7√x2 + 9

dx

10.∫

e√

x dx

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PHILOSOPHY ABOUT INTEGRATION

Philosophy about Integration

Definition. (Informal Definition) An elementary function is a function that can be builtup from familiar functions, like

• polynomials• trig functions• exponential and logarithmic functions

using familiar operations:

• addition• subtraction• multiplication• division• composition

Example. Give an example an elementary function. Make it as crazy as you can.

80


Question. Is it always true that the derivative of an elementary function is an elemen-tary function?

Question. Is it always true that the integral of an elementary function is an elementaryfunction?

81


Techniques of integration ... and their limitations.

82

§7.8 -IMPROPER INTEGRALS

§7.8 -Improper Integrals

Here are two examples of improper integrals:∫ ∞1

1x2

dx

and

∫ π2

0tan(x) dx

Question. What is so improper about them?

Definition. An integral is called improper if either

(Type I)

or,

(Type II)

or both.

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Type 1 Improper Integrals

To integrate over an infinite interval, we take the limit of the integrals over expandingfinite intervals

Example. Find∫ ∞

1

1x2

dx

Definition. The improper integral∫ ∞

af (x) dx is defined as ...

We say that∫ ∞

af (x) dx converges if ...

and diverges if ...

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Definition. Similarly, we define∫ b−∞

f (x) dx as ...

and say that∫ b−∞

f (x) dx converges if ...

and diverges ...

Example. Evaluate∫ −1−∞

1x

dx and determine if it converges or diverges.

85


Review. Which of the following are NOT improper integrals?

A.∫ ∞

1e−x dx

B.∫ 3

0

1x2

dx

C.∫ 5−5

ln |x| dx

D.∫ 0−∞

4x + 4

dx

E. They are all improper integrals.

Example. Evaluate∫ ∞

1

1√x

dx and determine if it converges or diverges.

86


Question. For what values of p > 0 does∫ ∞

1

1xp

dx converge?

87


Example. Find the area under the curve y = e3x−2 to the left of x = 2.

88


Type 2 Improper integrals

When the function we are integrating goes to infinity at one endpoint of an interval,we take a limit of integrals over expanding sub-intervals.

Definition. If f (x)→∞ or f (x)→ −∞ asx→ b−, then∫ b

af (x) dx =

Definition. If f (x)→∞ or f (x)→ −∞ asx→ a+, then∫ b

af (x) dx =

89


Example. Find the area under the curve y =x√

x2 − 1between the lines x = 1 and x = 2.

1 2 3 4 5

1.11.21.31.41.5

90


Review. True or False: If f (x) is continuous on (1, 2] and f (x) → ∞ as x → 1+, then∫ 21

f (x) dx diverges. (Hint: remember the pre-class video on Type 2 integrals.)

Example. Find∫ 10

1

4(x − 3)2 dx .

Note. Since4

(x − 3)2 blows up at x = 3, this integral must be computed as the sum oftwo indefinite integrals.

If you compute it without breaking it up YOU WILL GET THE WRONG ANSWER!

91


Question. For what values of p > 0 does∫ 1

0

1xp

dx converge?

92


Theorem. Comparison Theorem for Integrals: Suppose 0 ≤ g(x) ≤ f (x) on (a, b) (where a orb could be −∞ or∞).

(a) If∫ b

af (x) dx , then

∫ ba

g(x) dx also.

(b) If∫ b

ag(x) dx , then

∫ ba

f (x) dx also.

93


Example. Does∫ ∞

2

2 + sin(x)√x

dx converge or diverge?

94


Review. If 0 ≤ f (x) ≤ g(x) on the interval [a,∞), then which of the following are true?A. If

∫ ∞a f (x) dx converges, then

∫ ∞a g(x) dx converges.

B. If∫ ∞

a f (x) dx converges, then∫ ∞

a g(x) dx diverges.

C. If∫ ∞

a f (x) dx diverges, then∫ ∞

a g(x) dx converges.

D. If∫ ∞

a f (x) dx diverges, then∫ ∞

a g(x) dx diverges.

E. None of these are true.


1

cos(x) + 74x3 + 5x − 2 dx converge or diverge?

95



7

3x2 + 2x√x6 − 1

dx converge or diverge?

96



0e−x

2dx converge or diverge?

97


Question. What are some useful functions to compare to when using the comparisontest?

Question. True or False: Since −1x<

1x2

for 1 < x < ∞, and∫ ∞

1

1x2

dx converges, the

Comparison Theorem guarantees that∫ ∞

1−1

xdx also converges.

98


Comparison Test Practice Problems

Decide what function to compare to and whether the integral converges or diverges.

1.∫ ∞

1

1e5t + 2

dt

2.∫ ∞

2

√x2 − 1

x3 + 3x + 2dx

3.∫ ∞

1

x2

x2 + 4dx

4.∫ 2

0

√t + 2t2

dt

5.∫ ∞

5

6√t − 5

dt

6.∫ 5−1

cos(t) + 4√t + 1

dt Hint: do a u-

substitution.

7.∫ ∞

1

5ez + 2z

dz

8.∫ ∞

7

4 sin(x) + 5√x3 + x

dx

9.∫ ∞

7

x + 3√x4 − x

dx

10.∫ ∞

0

5√xex + 1

dx

99


Example. Find∫ ∞−∞

x cos(x2 + 1) dx

100


Question. True or False:∫ ∞−∞

f (x) dx = limt→∞

∫ t−t

f (x) dx

101

§11.1 - SEQUENCES

§11.1 - Sequences

Definition. A sequence is an ordered list of numbers.Example. 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 9, . . .

A sequence is often denoted {a1, a2, a3, . . .}, or {an}∞n=1, or {an}.

Example. For each sequence, write out the first three terms:

1.{

3n + 1(n + 2)!

}∞n=1

2.{

(−1)kk + 33k

}∞k=2

102

§11.1 - SEQUENCES

Definition. Sometimes, a sequence is defined with a recursive formula (a formula thatdescribes how to get the nth term from previous terms), such as

a1 = 2, an = 4 −1

an−1Example. Write out the first three terms of this recursive sequence.

Note. Sometimes it is possible to describe a sequence with either a recurvsive formulaor a ”closed-form”, non-recursive formula.

103

§11.1 - SEQUENCES

Example. Write a formula for the general term an, starting with n = 1.

A. {7, 10, 13, 16, 19, · · · }

Definition. An arithmetic sequence is a sequence for which consecutive terms havethe same common difference.

If a is the first term and d is the common difference, then the arithmetic sequence hasthe form:

(starting with n = 0)

An arithmetic sequence can also be written:

(with the index starting at n = 1.)

104

§11.1 - SEQUENCES

Example. For each sequence, write a formula for the general term an (start with n = 1or with n = 0).

B. {3, 0.3, 0.03, 0.003, 0.0003, · · · }

C.{

152 ,

754 ,

3758 ,

187516 , · · ·

}D. {3,−2, 43,−89, . . .}

Definition. A geometric sequence is a sequence for which consecutive terms have thesame common ratio.

If a is the first term and r is the common ratio, then a geometric sequence has the form:

(with the index starting at 0)

A geometric sequence can also be written:

(with the index starting at 1)

105

§11.1 - SEQUENCES

Example. For each sequence, write a formula for the general term an, starting withn = 1.

E. {−29, 416,− 825, 1636, . . .}

F. {−6, 5,−1, 4, 3, 7, 10, 17, . . .}

106

§11.1 - SEQUENCES

Definition. A sequence {an} is bounded above if

A sequence {an} is bounded below if:

Example. Which of these sequences are bounded?

A. {7, 10, 13, 16, 19, · · · }

B. {3, 0.3, 0.03, 0.003, 0.0003, · · · }

C.{

152 ,

754 ,

3758 ,

187516 , · · ·

}D. {3,−2, 43,−89, . . .}

107

§11.1 - SEQUENCES

Definition. A sequence {an} is increasing if

A sequence {an} is decreasing if

A sequence {an} is monotonic if it is increasing or decreasing.

Example. Which of these sequences are monotonic?

A. {7, 10, 13, 16, 19, · · · }

B. {3, 0.3, 0.03, 0.003, 0.0003, · · · }

C.{

152 ,

754 ,

3758 ,

187516 , · · ·

}D. {3,−2, 43,−89, . . .}

E. {−29, 416,− 825, 1636, . . .}

F. {−6, 5,−1, 4, 3, 7, 10, 17, . . .}

108

§11.1 - SEQUENCES

Definition. A sequence {an} converges if:

Otherwise, the sequence diverges. In other words, a sequence diverges if:

Example. Which of the following sequences converge?

A. {7, 10, 13, 16, 19, · · · }

B. {3, 0.3, 0.03, 0.003, 0.0003, · · · }

C.{

152 ,

754 ,

3758 ,

187516 , · · ·

}

109

§11.1 - SEQUENCES

Review. Give an example of a sequence that is

• monotonic and bounded

• monotonic but not bounded

• not monotonic but bounded

• not monotonic and not bounded

110

§11.1 - SEQUENCES

Example. Is the sequence{n − 5

n2

}∞n=1

monotonic? Bounded?

111

§11.1 - SEQUENCES

Review. A sequence {an} converges if:

Otherwise, the sequence diverges. In other words, a sequence diverges if:

Definition. More formally, we say limn→∞

an = L if:

We say limn→∞

an = ∞ if:

112

§11.1 - SEQUENCES

Example. Give an example of a sequence that

• converges

• diverges to∞ or −∞

• is bounded but still diverges

113

§11.1 - SEQUENCES

Review. Recall that a geometric sequence is a sequence that can be written in the form:

Here, r represents and a represents .

What is an example of a geometric sequence?

114

§11.1 - SEQUENCES

Example. Which of these are geometric sequences? Which of them converge?

•{

(−1)n4n5n+2

}∞n=0

•{5 · 0.5n

3n−1

}∞2

•{4/3, 2, 3,

92,274. . .

}

• {2,−4, 8,−16, 32,−64, . . .}

115

§11.1 - SEQUENCES

Question. For which values of a and r does {a · rn}∞n=0 converge?

116

§11.1 - SEQUENCES

The following are some techniques for proving that a sequence converges:

Trick 1: Recognize geometric sequences

Example. Does{

(−1)tet−13t+2

}∞t=0

converge or diverge?

117

§11.1 - SEQUENCES

Trick 2: Suppose an = f (n) for some function f , where n = 1, 2, 3, . . .. If limx→∞

f (x) = Lthen lim

n→∞an = L.

So ... replace an with f (x) and use l’Hospital’s Rule or other tricks from Calculus 1 toshow that lim

x→∞f (x) exists.

Example. {an} ={

ln(1 + 2en)n

}

118

§11.1 - SEQUENCES

Trick 3: Use the Squeeze Theorem: trap the sequence between two simpler sequencesthat converge to the same limit.

Example. {an} ={

cos(n) + sin(n)n2/3

}

119

§11.1 - SEQUENCES

Example. 0.1, 0.12, 0.123, 0.1234, . . . , 0.12345678910, 0.1234567891011, 0.123456789101112,. . .

Trick 4: If {an} is and , then it converges.

120

§11.1 - SEQUENCES

Trick 5: Use the Limit Laws

The usual limit laws about addition, subtractions, etc. hold for sequences as well asfor functions. (See p. 693 in textbook.)

For example, if limn→∞

an = L and limn→∞

bn = M, then

limn→∞

(an + bn) =

limn→∞

(anbn) =

limn→∞

(can) = (c is a constant)

Example.{

k2

2k2 − k +4 · πk

6k

}∞k=3

121

§11.1 - SEQUENCES

True or False:

1. If ak converges, then so does |ak|.2. If |ak| converges, then so does ak.3. If ak converges to 0, then so does |ak|.4. If |ak| converges to 0, then so does ak.

True or False:

1. Suppose an = f (n) for some function f , where n = 1, 2, 3, . . .. If limx→∞

f (x) = L thenlimn→∞

an = L.

2. Suppose an = f (n) for some function f . If limn→∞

an = L, then limx→∞

f (x) = L.

122

§11.1 - SEQUENCES

Additional problems if additional time:

Do the following sequences converge or diverge? Justify your answer.

1.{

cos( j)ln( j + 1)

}∞j=1

2.{

(−1)t4t−132t

}∞t=3

3. 3√kln(k)

∞k=2

4.{3n

n!

}∞n=1

5.{n!

3n

}∞n=1

123

§11.2 - SERIES

§11.2 - Series

Definition. For any sequence {an}∞n=1, the sum of its terms a1 + a2 + a3 + · · · is a series.Often this series is written as

∞∑n=1

an

Example. Consider the sequence{

12n

}∞n=1

. If we add together all the terms, we get theseries:

∞∑n=1

12n

=

What does it mean to add up infinitely many numbers?

124

§11.2 - SERIES

Definition. The partial sums of a series∞∑

n=1

an are defined as the sequence {sn}∞n=1, where

s1 =

s2 =

s3 =

sn =

Definition. The series∞∑

n=1

an is said to converge if :

Otherwise, the series diverges.

Note. Associated with any series∞∑

n=1

an, there are actually two sequences of interest:

1.

2.

125

§11.2 - SERIES

Example. For the series∞∑

n=1

1n2 + n

, write out the first 4 terms and the first 4 partial

sums. Does the series appear to converge?

126

§11.2 - SERIES

Review. Using your calculator, Excel, or any other methods, compute several partialsums for each of the following series and make conjectures about which series convergeand which diverge.

A.∞∑

k=1

14k

B.∞∑j=1

(−1) j

C.∞∑

t=1

2t − 15t + 2

D.∞∑

n=2

3n2 − 1

E.∞∑

n=1

1n

127

§11.2 - SERIES

Tricks for determining when series converge:

Trick 1: If the sequence of terms an do not converge to 0, then the series∑

an has nohope of converging.Theorem. (The Divergence Test) If

then the series∞∑

n=1

an diverges.

Example. .

Example. .

Note. If the sequence of terms an do converge to 0, then the series∑

an.

128

§11.2 - SERIES

Trick 2: Recognize geometric series.

Recall that the geometric sequence {arn−1}∞n=1 converges to 0 when , convergesto when and diverges when .

Question. For what values of r does the geometric series∞∑

n=1

arn−1 converge?

129

§11.2 - SERIES

Conclusion: The geometric series∞∑

n=1

arn−1 converges to when .

The geometric series∞∑

n=1

arn−1 diverges when .

Example.∞∑

i=2

5(−2)i33i−4

130

§11.2 - SERIES

Trick 3: Recognize telescoping series.

Example.∞∑

k=2

ln(

kk + 1

)

131

§11.2 - SERIES

Example.∞∑

n=2

3n2 − 1

132

§11.2 - SERIES

Trick 4: Recognize the Harmonic Series:

Question. Does the Harmonic Series converge or diverge?

133

§11.2 - SERIES

Trick 5: Use Limit Laws.

Fact 0.1. If∞∑

n=1

an = A and∞∑

n=1

bn = B, then

∞∑n=1

an + bn =

∞∑n=1

an − bn =

∞∑n=1

c · an =

where c is a constant.

Be careful! ∞∑n=1

an · bn =

∞∑n=1

anbn

=

134

§11.2 - SERIES

Example. Does the series converge or diverge?∞∑

n=1

4 · 5n − 5 · 4n6n

135

§11.2 - SERIES

Question. True or False: If∞∑

n=1

an converges, then so does∞∑

n=5

an.

Question. True or False: If∞∑

n=5

an converges, then so does∞∑

n=1

an.

136

§11.3 - THE INTEGRAL TEST

§11.3 - The Integral Test

Example. Does this series converge or or diverge?

∞∑n=1

1n2

137


The series∞∑

n=1

1n2

is closely related to the improper integral∫ ∞

1

1x2

dx .

138


Example. Does this series converge or or diverge?

∞∑n=1

1√x

139


Theorem. (The Integral Test) Suppose f is a continuous, positive, decreasing functionon [1,∞) and an = f (n). Then

1. If∫ ∞

1f (x) dx converges, then

∞∑n=1

an converges.

2. If∫ ∞

1f (x) dx diverges, then

∞∑n=1

an diverges.

140


Example. Does∞∑

n=1

ln nn


141


Review. We know that∫ ∞

1

1x2

dx converges to 1. Which of the following are true?

A.∞∑

n=1

1n2

converges.

B.∞∑

n=1

1n2

= 1.

C. Both of the above.

D. None of the above.

142


Example. Does∞∑

n=1

nen


143


Example. Does the following series converge or diverge?

15

+18

+1

11+

114

+ · · ·

144


Question. For what values of p does the p-series∞∑

n=1

1np

converge?

145


Bounding the Error

Definition. If∞∑

n=1

an converges, and sn is the nth partial sum, then for large enough n, sn

is a good approximation to the sum s∞ =∞∑

k=1

ak. Define Rn be the error, or remainder:

Rn =

If an = f (n) for a continuous, positive, decreasing function f (x), use the picture to puta bound on how big Rn can be.

≤ Rn ≤

This is called the Remainder Estimate for the Integral Test

146


Example. (a) Put a bound on the remainder when you use the first three terms to

approximate∞∑

n=1

6n2

.

(b) How many terms are needed to approximate the sum to within 3 decimal places?

147


Question. Which of the following are always true?

1. Suppose f is a continuous, positive, decreasing function on [1,∞) and for n ≥ 1,

an = f (n). Then∞∑

n=1

an converges, if and only if∫ ∞

1f (x) dx converges.

2. Suppose f is a continuous, positive, decreasing function on [5,∞) and for n ≥ 5,

an = f (n). Then∞∑

n=1

an converges if and only if∫ ∞

5f (x) dx.

3. Suppose f is a continuous, positive function on [1,∞) and for n ≥ 1, an = f (n).

Then∞∑

n=1

an converges if and only if∫ ∞

1f (x) dx converges.

148

§11.4 - COMPARISON TESTS FOR SERIES

§11.4 - Comparison Tests for Series

Theorem. (The Comparison Test for Series) Suppose that∑∞

n=1 an and∑∞

n=1 bn are series and0 ≤ an ≤ bn for all n.

1. If converges, then converges.

2. If diverges, then diverges.

Note. The following series are especially handy to compare to when using the com-parison test.

1. which converges when

2. which converges when

149


Example. Does∞∑

n=1

3n

5n + n2converge or diverge?

150


Theorem. (The Limit Comparison Test) Suppose∑

an and∑

bn are series with positive terms.If

limn→∞

anbn

= L

where L is a finite number and L > 0, then either both series converge or both diverge.

Example. Does∞∑

n=1

3n

5n − n2 converge or diverge?

151


Review. Suppose∑∞ an and ∑∞ bn are series whose terms are ≥ 0. Which of the

following will allow us to conclude that∑∞ bn diverges?

A. an ≥ bn for all n and∑∞ an converges.

B. an ≥ bn for all n and∑∞ an diverges.

C. an ≤ bn for all n and∑∞ an converges.

D. an ≤ bn for all n and∑∞ an diverges.

Review. The (Ordinary) Comparison Test for Series: Suppose that∑∞

n=1 an and∑∞

n=1 bnare series with positive terms and 0 ≤ an ≤ bn for all n.

1. If converges, then converges.

2. If diverges, then diverges.

152


Review. Suppose∑∞ an and ∑∞ bn are series with positive terms. Which of the follow-

ing will allow us to conclude that∑∞ bn converges? (More than one answer may be

correct.)

A. limn→∞

an = limn→∞

bn and∞∑

an converges.

B. limn→∞

anbn

= 0 and∑∞ an converges.

C. limn→∞

anbn

=13

and∑∞ an converges.

D. limn→∞

anbn

= 5 and∑∞ an converges.

Review. The Limit Comparison Test: Suppose∑

an and∑

bn are series with positiveterms. If

limn→∞

anbn

= L

where L ,then:

153


Advice on the Comparison Theorems:Question. What series are especially handy to compare to when using the comparisontest?

Question. How to decide whether to use the Ordinary Comparison Test or the LimitComparison Test?

154


Example. Decide if the series converges or diverges.

∞∑n=1

3n − 5√n3 + 2n

155


Example. Decide if∞∑

n=3

n sin2(n)n3 + 7n

converges or diverges.

156


Example. Decide if∞∑

n=3

n sin2(n)n3 − 7n converges or diverges.

157


Question. True or False: If limn→∞

anbn

= 0, then the series∑

an and∑

bn have the sameconvergence status.

Can anything be concluded if limn→∞

anbn

= 0?

158


Question. Find the error: Consider the two series∞∑

n=1

an = (−1) + (−2) + (−3) + (−4) + (−5) + (−6) . . .

and ∞∑n=1

bn = 2 + (−1) + (1/2) + (−1/4) + (1/8) + (−1/16) + . . .

Note that∑∞

n=1 is a geometric series with ratio r = −1/2.Since an ≤ bn for all n, and

∑bn converges,

∑an also converges, by the Ordinary

Comparison Test.

159


Note. Review of the convergence tests for series so far:

1.

2.

3.

4.

5.

6.

160

SECTION 11.5 - ALTERNATING SERIES

Section 11.5 - Alternating Series

Definition. An alternating series is a series whose terms are alternately positive andnegative. It is often written as

∞∑k=1

(−1)k−1bk

where the bk are positive numbers.Example. (The Alternating Harmonic Series)

161


Does the Alternating Harmonic Series converge? Hint: look at ”even” partial sumsand ”odd” partial sums separately.

162


Theorem. (Alternating Series Test) If the series∞∑

n=1

(−1)n−1bn = b1 − b2 + b3 − b4 . . .

satisfies:

1.

2.

3.

then the series is convergent.

Example. Which of these series are guaranteed to converge by the Alternating SeriesTest?

A. 12 − 23 + 34 − 45 + 56 − 67 + · · ·B. 5√

2− 5√

3+ 5√

4− 5√

5+ 5√

6− 5√

7+ · · ·

C. 22 − 12 + 23 − 13 + 24 − 14 + 25 − 15 + · · ·D. 18 − 14 + 127 − 19 + 164 − 116 + 1125 − 125 + · · ·

163


Note. Why is the condition limn→∞

bn = 0 necessary?

Note. Why is the condition bn+1 ≤ bn for all large n necessary?

164


Example. Does the series converge or diverge?

∞∑n=1

(−1)n n2

n3 − 2

165



∞∑k=1

(−1)k(1 + k)1/k

166


Bounding the Remainder

For the same type of series:

• series is alternating• limn→∞ bn = 0• bn+1 ≤ bn

We want to put a bound on the remainder. Call the sum of the infinite series S and thenth partial sum Sn.

1. Write an equation for the nth remainder Rn.

2. Find an upper bound on |Rn|:|Rn| ≤

167


Example. Consider the series −14 + 19 − 116 + 125 − · · ·If we add up the first 6 terms of this series, what is true about the remainder? (PollEv)

A. positive and < 0.01

B. positive and < 0.02

C. positive and < 0.05

D. negative with absolute value < 0.01

E. negative with absolute value < 0.02

F. negative with absolute value < 0.05

G. none of these.

168


Example. How many terms of the series

−14

+19− 1

16+

125− · · ·

do we need to add up to approximate the limit to within 0.01?

169

§11.6 - RATIO AND ROOT TESTS

§11.6 - Ratio and Root Tests

Definition. A series∑

an is called absolutely convergent if

Example. Which of these series are convergent? Which are absolutely convergent ?

1.∞∑

m=0

(−0.8)m convergent abs. convergent

2.∞∑

k=1

1√k

convergent abs. convergent

3.∞∑j=5

(−1) j1j

convergent abs. convergent

170


Question. Is it possible to have a series that is convergent but not absolutely conver-gent?

Definition. A series∑

an is called conditionally convergent if

Question. Is it possible to have a series that is absolutely convergent but not conver-gent?

171



∞∑n=2

cos(n) + sin(n)n3

172


Recall: for a geometric series∑

arn

Theorem. (The Ratio Test) For a series∑

an :

1. If limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = L < 1, then ∞∑

n=1

an is .

2. If limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = L > 1 or limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = ∞, then ∞∑

n=1

an is .

3. If limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = 1, then ∞∑

n=1

an .

173


Example. Apply the ratio test to∞∑

n=1

n2(−10)nn!

174


Review. Which of the following statements are true about a series∞∑

an?

A. If the series is absolutely convergent, then it is convergent.

B. If the series is convergent, then it is absolutely convergent.

C. Both are true.

D. None of these statements are true.

Question. Which of the following Venn Diagrams represents the relationship betweenconvergence, absolute convergence, and conditional convergence?

175


Review. In which of these situations can we conclude that the series∞∑

an converges?

A. limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = 0

B. limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = 0.3

C. limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = 1

D. limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = 17

E. limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = ∞

Review. (The Ratio Test) For a series∑

an :

1. If limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = L < 1, then ∞∑

n=1

an is .

2. If limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = L > 1 or limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = ∞, then ∞∑

n=1

an is .

3. If limn→∞

∣∣∣∣∣an+1an∣∣∣∣∣ = 1 or DNE , then .

176


Example. Apply the ratio test to∞∑

n=1

(1.1)n

(2n)!

177


Example. Apply the ratio test to the series∞∑

n=2

3n2 − n

178


Extra Example. Apply the ratio test to the series

a1 = 1, an =sin n

nan−1

179


Theorem. (The Root Test)

1. If limn→∞

n√|an| = L > 1 or lim

n→∞n√|an| = ∞, then

∞∑n=1

an .

2. If limn→∞

n√|an| = L < 1, then

∞∑n=1

an .

3. If limn→∞

n√|an| = 1, then

∞∑n=1

an .

Example. Determine the convergence of∞∑

n=1

5n

nn

180


RearrangementsDefinition. A rearrangement of a series

∑an is a series obtained by rearranging its

terms.Fact. If

∑an is absolutely convergent with sum s, then any rearrangement of

∑an also

has sum s.

But if∑

an is any conditionally convergent series, then it can be rearranged to give adifferent sum.Example. Find a way to rearrange the Alternating Harmonic Series so that the rear-rangement diverges.

Example. Find a way to rearrange the Alternating Harmonic Series so that the rear-rangement sums to 2.

181

§11.7 - STRATEGY FOR CONVERGENCE TESTS FOR SERIES

§11.7 - Strategy for Convergence Tests for Series

List as many convergence tests as you can. What conditions have to be satisfied?

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

182


Question. The limit comparison test and the ratio test both involve ratios. How arethey different?

183


Example. Which convergence test would you use for each of these examples? Carryout the convergence test if you have time.

1.∞∑

n=1

2n

n3

2.∞∑

n=1

(−1)n ln nn + 3

3.∞∑

n=1

13√

n2 + 6n

4.∞∑

n=1

1n!− 1

2n

5.∞∑

n=1

n2

en2

6.∞∑

n=1

3n ln n

184

§11.8 - POWER SERIES

§11.8 - Power Series

Informally, a power series is a series with a variable in it (often ”x”), that looks like apolynomial with infinitely many terms.

Example.∞∑

n=0

(2n + 1)xn

3n−1= 3 + 3x +

5x2

3+

7x3

9+

9x4

27+

11x5

81+ · · ·

is a power series.

Example.∞∑

n=0

(5n)(x − 6)nn!

= 1 + 5(x − 6) + 52(x − 6)2

2!+

53(x − 6)33!

+54(x − 6)4

4!+

55(x − 6)55!

+ · · ·

is a power series centered at 6.

185


Definition. A power series centered at a is a series of the form∞∑

n=0

cn(x − a)n =

where x is a variable, and the cn’s are constants called coefficients, and a is also aconstant called the center .

Definition. A power series centered at zero is a series of the form∞∑

n=0

cnxn =

186


Example. For what values of x does the power series∞∑

n=0

n! (x − 3)n converge?

187



n=0

(−2)n(x + 4)nn!

converge?

188



n=1

(−5x + 2)nn

converge?

189


Review. Which of the following are power series?

A.(x + 1)

3+

(x + 1)2

6+

(x + 1)3

9+

(x + 1)4

12+ · · ·

B.1x2

+1x

+ 1 + x + x2 + x3 + x4 + · · ·

C. 1 + 3 + 32 + 33 + 34 + · · ·D. None of these.

190



n=1

nn(7 + 3x)n converge?

Hint: limn→∞

(1 +

1n

)n= e.

191



n=0

(−5)n(2x − 3)n√3n + 1

converge?

192



n=0

x2n

(2n)!converge?

193


Theorem. For a given power series∞∑

n=0

cn(x − a)n, there are only three possibilities for conver-gence:

1.

2.

3.Definition. The radius of convergence is

1.

2.

3.Definition. The interval of convergence is the interval of all x-values for which thepower series converges.

1.

2.

3.

194


Question. If the interval of convergence of a power series has length 6, then the radiusof convergence of the power series is:

Question. Which of the following could NOT be the interval of convergence for apower series?

A. (−∞,∞)B. (−4, 1]C. (0,∞)D. [92,

1003 ]

Question. If the series∑∞

n=1 cn5n converges, which of the following definitely con-verges?

A.∑∞

n=1 cn(−3)n

B.∑∞

n=1 cn(−5)n

C.∑∞

n=1 cn(−7)n

D. None of these.

195


Extra Example. Find the radius of convergence and the interval of convergence for∞∑

n=1

(−4)n(x − 8)2nn

196

§11.9 - APPROXIMATING FUNCTIONS WITH POWER SERIES

§11.9 - Approximating functions with power series

We can think of power series as functions.

Example. Consider f (x) =∞∑

n=0

xn =

1. What is f (13)?

2. What is the domain of f (x)?

3. What is a closed form expression for f (x)?

4. What is the domain for the closed form expression?

197


We can think of the partial sums of∞∑

n=0

xn as a way to approximate the function 11−x

with polynomials:

s0 =

s1 =

s2 =

s3 =

sn =

198


Example. Express2

x − 3 as a power series and find the interval of convergence.

199


Example. Find a power series representation ofx

1 + 5x2

200


Review. 11−x can be represented by the power series:

Question. 11−x is equal to its power series:

A. when x , 1

B. when x < 1

C. when −1 < x < 1D. for all real numbers

E. It is never exactly equal to its power series, only approximately equal.

201


Example. Express each of the following functions with a power series.

1.1

1 − x4

2.1

1 + x4

3.x3

1 + x4

202


Example. Find a power series representation of f (x) = 32+5x. Find its radius of conver-gence.

203


Differentiation and IntegrationRecall how to differentiate and integrate polynomials:

ddx

[5 + 3(x − 2) + 4(x − 2)2 + 8(x − 2)3] =...∫

5 + 3(x − 2) + 4(x − 2)2 + 8(x − 2)3 dx =

Power series are also very easy to differentiate and integrate!

Theorem. If the power series

f (x) =∞∑

n=0

cn(x − a)n = c0 + c1(x − a) + c2(x − a)2 + c3(x − a)3 + c4(x − a)4 · · ·

has a radius of convergence R > 0, then f (x) is differentiable on the interval (a−R, a + R) and(i) f ′(x) =

(ii)∫

f (x) dx =

The radius of convergence of the power series in (i) and (ii) are both R.

204


Example. Find a power series representation for ln |x + 2| and find its radius of conver-gence.

205


Example. Find a power series representation for( 14x − 1

)2and find its radius of con-

vergence.

206


Example. Find a power series representation for∫

x8 + x3

dx and use it to approximate∫ 10

x8 + x3

dx, accurate to two decimal places.

207


Summary:

• We started by representing the function 11 − x with the power series

∞∑n=0

1 + x + x2 + x3 + · · ·

• We used the equation∞∑

n=0

11 − x = 1 + x + x

2 + x3 + · · · as a template to find the

power series for many other related functions, by:

–

–

–

–

• These same techniques can be used with other templates to build new power seriesout of old ones.

208

§11.10 - TAYLOR SERIES

§11.10 - Taylor Series

A General Method for Representing Functions as Power Series

Idea: Approximate a function with a polynomial.

Suppose we want to approximate a function f (x) near x = 0. Assume that f’s derivative,second derivative, third derivative, etc all exist at x = 0.

209


Review. Let f (x) be a function whose derivatives all exist near 5. Suppose that f (x) can

be represented with a power series: f (x) =∞∑

n=0

cn(x − 5)n on some interval of x-values

around 5. Write an expression for f (4)(5) in terms of the coefficients cn.

210


Note. For f (x) =∞∑

n=0

cn(x − a)n,

f (n)(a) =

Therefore,

cn =

Theorem. If f (x) has a power series expansion centered at a, then it must be of the form:

f (x) =

Definition. This power series is called the of the function f (x)centered at a.

We use the conventions that:

• f (0)(a) means• 0! =• (x − a)0 =

211


Definition. The power series∞∑

n=0

f (n)(0)n!

xn = f (0) +f ′(0)1!

x +f ′′(0)

2!x2 +

f ′′′(0)3!

x3 + · · ·

is called the for f (x).

Question. What is the difference between a Taylor series and a Maclaurin series?

Definition. The partial sums of the Taylor series are called the Taylor polynomials.

The nth degree Taylor polynomial is written:

• For today, we will assume that all of our Taylor series actually converge to thefunctions they are made from, on their intervals of convergence.

• We will see later that it is actually possible for a Taylor series NOT to converge tothe function it is made from, but we won’t come across this pathology very often.

212


Example. A. Find the Taylor series for f (x) = ln x centered at a = 2.

B. What is the radius of convergence?

C. If T(x) is the Taylor series for f (x) = ln x, find T(3)(2) and compare it to f (3)(2).

f (x)T3(x)

T9(x)

T6(x)-2 2 4 6

-5

5

213


Example. Find the Maclaurin series for f (x) = sin(x) and g(x) = cos(x). Find the radiusof convergence.

214


Example. Find the Maclaurin series for f (x) = ex. What is the radius of convergence?

215


Example. Use a Taylor series to evaluate limx→0

e−x2 − 1 + x2x4

216


Example. Use Taylor series to prove L’Hospital’s Rule.

217


Example. Find the Maclaurin series for g(x) = eix, where i =√−1.

218


Example. 1. Find a power series representation for∫

e−x22 dx.

2. Use the first three terms of your power series to estimate1√2π

∫ 1−1

e−x22 dx.

What does this number represent?

219


Additional MacLaurin series are listed in this table.

Example. Use the MacLaurin series for arctan(x) to show that

1 − 1/3 + 1/5 − 1/7 + · · · = π4

220


Example. Use the MacLaurin series for f (x) = ln x + 1 at x = 1 to calculate the limit ofthe alternating harmonic series.

221


Example. Find the Taylor series for f (x) = (1 + x)π centered at x = 0.

222


Definition. The expressionk(k − 1)(k − 2) . . . (k − n + 1)

n!is written as ,

pronounced , and is also called a .

Definition. The binomial series is the Maclaurin series for (1 + x)k, where k is any realnumber. That is, the binomial series is the series:

(1 + x)k =

This series converges when .

Example. Find the Maclaurin series for 1√1+x3

.

223


Extra Example. If P(x) =∞∑

n=0

5n!

(x − 2)n = 5 + 51!

(x − 2) + 52!

(x − 2)2 + · · ·, find P′′′(2).

A. 5

B.52!

C.53!

D.5 · 23

3!E. None of these.Extra Example. Find a power series P(x) such that P(n)(5) = n for all n ≥ 0.

A.∞∑

n=1

n(x − 5)n

B.∞∑

n=1

(x − 5)n(n − 1)!

C.∞∑

n=1

(x − 5)nn!

D. None of these

224


Summary: What are Taylor Series good for?

225

§11.10 AND §11.11 - TAYLOR SERIES THEORY AND REMAINDERS

§11.10 and §11.11 - Taylor Series Theory and Remainders

Taylor Series TheoryQuestion. Does the Taylor series always converge to the function it’s made from?

No. It is not true that every (infinitely differentiable) function can be represented asthe sum of its Taylor series.

• Sometimes the radius of convergence is 0.• Sometimes the radius of convergence is large or even infinite, so the Taylor series

converges ... but to the wrong function!Example.

g(x) =

e−1/x2, if x , 0

0, if x = 0

226


Definition. For a function f (x) and its Taylor series T(x), the remainder is written

Rn(x) = f (x) − Tn(x) =

Question. How is this definition of remainder similar to our previous definition ofremainder? How is it different?

Theorem. The Taylor series for f (x) converges to f (x) in an interval around a if and only iflimn→∞

Rn(x) = 0 for all x in this interval.

227


Theorem. (Taylor’s Inequality) If | f (n+1)(x)| ≤ M for |x − a| ≤ d, then the remainder Rn(x) ofthe Taylor series satisfies the inequality

|Rn(x)| ≤M

(n + 1)!|x − a|n+1

for |x − a| ≤ d.

228


Theorem. (Practical convergence condition) If there is a number M such that | f (n)(x)| < Mfor all x with |x − a| < d, and for all n, then the Taylor series for f (x) converges to f (x) for|x − a| < d.

Note. If this practical convergence condition does not hold, the Taylor series may ormay not converge to f (x).

229


Example. Prove that the Taylor Series for sin(x) converges to sin(x).

230


Review. How is the remainder defined for Taylor series?

A. Rn(x) =∞∑

k=0

f (k)(x)k!

(x − a)k −n∑

k=0

f (k)(x)k!

(x − a)k

B. Rn(x) = f (x) −n∑

k=0

f (k)(x)k!

(x − a)k

C. Both of these.

D. Neither of these.

Review. (Taylor’s Inequality) If | f (n+1)(x)| ≤ M for |x − a| ≤ d, then the remainder Rn(x)of the Taylor series satisfies the inequality

for |x − a| ≤ d.

231


Review. (Practical convergence condition) If there is a number M such that

for all x with |x − a| < d, and for all n,

then the Taylor series for f (x) converges to f (x) for |x − a| < d.

232


Example. Prove that the Taylor series for ex converges to ex.

233


Example. Approximate f (x) = cos(x) by a Taylor polynomial of degree 4.

1. Estimate the accuracy of the approximation when x is in the interval [0, π/2]

2. For what values of x is the approximation accurate to within 3 decimal places?

3. Check out the approximation graphically.

234


Example. Approximate f (x) = ex/3 by a Taylor polynomial of degree 2 at a = 0. Estimatethe accuracy of the approximation when x is in the interval [−0.5, 0.5].

235

§8.1 - ARCLENGTH

§8.1 - Arclength

Example. Find the length of this curve.

236

§8.1 - ARCLENGTH

Note. In general, it is possible to approximate the length of a curve y = f (x) betweenx = a and x = b by dividing it up into n small pieces and approximate each curvedpiece with a line segment.

Arclength is given by the formula:

237

§8.1 - ARCLENGTH

Example. Find the arclength of y = x3/2 between x = 1 and x = 4.

238

§8.1 - ARCLENGTH

Example. Find the arc length of the curve x = y2 between x = 0 and x = 3.

239

§8.1 - ARCLENGTH

Example. Find a function a(t) that give the length of the curve y = ex+e−x

2 between x = 0and x = t.

240

§8.1 - ARCLENGTH

Note. Although arc length integrals are usually straightforward to set up, the squareroot sign makes them notoriously difficult to evaluate, and sometimes impossible toevaluate.

241

§10.1 - PARAMETRIC EQUATIONS

§10.1 - Parametric Equations

Definition. A cartesian equation for a curve is an equation in terms of x and y only.Definition. Parametric equations for a curve give both x and y as functions of a thirdvariable (usually t). The third variable is called the parameter.Example. Graph x = 1 − 2t, y = t2 + 4

t x y-2 5 8-1 3 50

Find a Cartesian equation for this curve.

242


Example. Plot each curve and find a Cartesian equation:

1. x = cos(t), y = sin(t), for 0 ≤ t ≤ 2π2. x = cos(−2t), y = sin(−2t), for 0 ≤ t ≤ 2π3. x = cos2(t), y = cos(t)

243


Example. Write the following in parametric equations:

1. y =√

x2 − x for x ≤ 0 and x ≥ 1

2. 25x2 + 36y2 = 900

244


Example. Describe a circle with radius r and center (h, k):

a) with a Cartesian equation

b) with parametric equations

245


Which of the following graphs represents the graph of the parametric equations x =cos t, y = sin t. (The axes on the graphs are the x and y axes.)

1.

2.

3.

246


Review. What is the equation for a circle of radius 8 centered at the point (5, -2)

1. in Cartesian coordinates (i.e. in terms of x and y only) ?

2. in parametric equations (i.e. in terms of a third variable t) ?

247


Example. Find a Cartesian equation for the curve.

1. x = 5√

t, y = 3 + t2

2. x = et, y = e−t

3. x = 5 cos(t) + 3, y = 2 sin(t) − 7

248


Example. Find parametric equations for the curve.

1. x = −y2 − 6y − 9

2. 4x2 + 25y2 = 100

3. 4(x − 2)2 + 25(y + 1)2 = 100

249


Example. Find parametric equations for a line through the points (2, 5) and (6, 8).

1. any way you want.

2. so that the line is at (2, 5) when t = 0 and at (6, 8) when t = 1.

250


Example. Lissajous figure: x = sin(t), y = sin(2t)

251


Example. Use the graphs of x = f (t) and y = g(t) to sketch a graph of y in terms of x.

252


Extra Example. A double ferris wheel has a big stick of radius 5 meters that rotatescounterclockwise 1 time per minute and a small wheel on each end of the stick ofradius 2 meters that rotates clockwise 6 times per minute.

1. Find parametric equations to describe the position of a rider. Hint: first find the x-and y- coordinates of the center of the small wheel. Then find how far the rider isin the x- and y- direction from this center.

2. Use a calculator or computer to graph the motion of the rider.

253

§10.2 CALCULUS USING PARAMETRIC EQUATIONS

§10.2 Calculus using Parametric Equations

ARC LENGTHExample. Find the length of this curve.

254


Note. In general, it is possible to approximate the length of a curve x = f (t), y = g(t)between t = a and t = b by dividing it up into n small pieces and approximating eachcurved piece with a line segment.

Arc length is given by the formula:

255


Set up an integral to express the arclength of the Lissajous figure

x = cos(t), y = sin(2t)

.

256


Review. The length of a parametric curve y = f (t), y = g(t) from t = a to t = b is givenby:

Example. Find the exact length of the curve x = cos(t) + t sin(t), y = sin(t) − t cos(t),from the point (1, 0) to the point (−1, π).

257


Example. Write down an expression for the arc length of a curve given in Cartesiancoordinates: y = f (x).

Example. Find the arc length of the curve y = 12 ln(x) − x2

4 from x = 1 to x = 3.

258


SURFACE AREA

To find the surface area of a surface of revolution, imagine approximating it with piecesof cones.

We will need a formula for the area of a piece of a cone.

The area of this piece of a cone is

A = 2πr`

where r =r1 + r2

2is the average radius and

` is the length along the slant.

259


Use the formula for the area of a piece of cone A = 2πr` to derive a formula for surfacearea of the surface formed by rotating a curve in parametric equations around thex-axis.

260


Example. Prove that the surface area of a sphere is 4πr2

261


Example. The infinite hotel:

1. You are hired to paint the interior surface of an infinite hotel which is shaped likethe curve y = 1x with x ≥ 1, rotated around the x-axis. How much paint will youneed? (Assume that a liter of paint covers 1 square meter of surface area, and xand y are in meters.)

2. Your co-worker wants to save time and just fill the hotel with paint to cover allthe walls and then suck out the excess paint. How much paint is needed for yourco-worker’s scheme?

262

§10.3 POLAR COORDINATES

§10.3 Polar Coordinates

Cartesian coordinates: (x, y)Polar coordinates: (r, θ), where r is:. and θ is:

Example. Plot the points, given in polar coordinates.

1. (8,−2π3 )

2. (5, 3π)

3. (−12, π4 )

Note. A negative angle means to go clockwise from the positive x-axis. A negativeradius means jump to the other side of the origin, that is, (−r, θ) means the same pointas (r, θ + π)

263


Note. To convert between polar and Cartesian coordinates, note that:

• x =• y =• r =• tanθ =

Example. Convert (5,−π6 ) from polar to Cartesian coordinates.

Example. Convert (−1,−1) from Cartesian to polar coordinates.

264


Review. Convert the point P = (4, −2π3 ), which is in polar coordinates, to Cartesiancoordinates.

A. (12,√

32 )

B. (−12,−√

32 )

C. (−2, 2√

3)

D. (−2,−2√

3)

E. None of these.

Review. Convert the point P = (−√

3, 3), which is in Cartesian coordinates, to polarcoordinates. (More than one answer may be correct.)

A. (1, π3 )

B. (2√

3, π3 )

C. (2√

3, 2π3 )

D. (−2√

3, −π3 )

E. None of these.

265


Example. Plot the following curves and rewrite the first three using Cartesian coordi-nates.

A. r = 7 C. r = 12 cos(θ)B. θ = 1 D. r = 6 + 6 cos(θ) (an example of a limacon)

266


Example. Describe the regions using polar coordinates.

267

§10.4 AREA IN POLAR COORDINATES

§10.4 Area in polar coordinates

Goal: Find a formula for the area of a region whose boundary is given by a polarequation r = f (θ).

Step 2: Find a formula for a sector of a circle.

268


Step 2: Divide our polar region with boundary r = f (θ) into slivers ∆A that areapproximately sectors of circles.

Step 3: Approximate the total area with a Riemann sum.

Step 4: Take the limit of the Riemann sum to get an integral.

269


Example. Find the area inside one leaf of the flower r = sin(2θ)

270


Extra Example. Find the area of the region that lies inside both flowers: r = sin(2θ)and r = cos(2θ)

271

§4.4 - L’HOSPITAL’S RULE

§4.4 - L’Hospital’s Rule

Example. limx→∞

ln(x)3√x

Example. limx→0+

sin(x) ln(x)

272


Example. limx→5+

ex

x − 5

Example. limx→0+

xx

273


Example. limx→∞

ln(x2 − 1) − ln(x5 − 1)

Tips for using L’Hopital’s Rule:

274


Form Example What to do

275

§9.1, 9.2, 9.3 - DIFFERENTIAL EQUATIONS

§9.1, 9.2, 9.3 - Differential Equations

Differential equations are equations that involve functions and their derivatives. Forexample,

1. dydx =√

x

2. y′ = 1 + y2

3. d2y

dx2 = −4y4. y′ = x + y

Solving a differential equation means to find all functions y = f (x) that satisfy it.Sometimes it is useful to find a particular solution, with a given initial condition, suchas y(2) = 5.

276


Example. dydx =√

x

1. Solve this differential equation.

2. How do you know you have found all solutions?

277


Example. y′ = 1 + y2

1. Verify that y = tan(x) is a solution to this equation.

2. Is y = tan(x) + 3 a solution?

3. Is y = 3 tan(x) a solution?

4. Is y = tan(x + 3) a solution?

5. Find a solution that satisfies the initial condition y(0) = 1.

278


Example. ”Separate” the differential equation by moving all y’s to the left side and allx’s to the right side, to find all solutions to the equation

y′ = 1 + y2

279


Example. Find a solution the equation

dydx

= xy2

1. with the initial condition y(0) = 4.

2. with the initial condition y(1) = 0.

280


Definition. An equation of the form

dydx

= g(x) f (y)

is called a separable differential equation.

Equivalently, an equation of the form

dydx

=g(x)h(y)

is called a separable differential equation. Here, f (y) =1

h(y)

Separable differentiable equations can be solved by moving expressions with y’s inthem to the left side of the equals sign and expressions with x’s in them on the rightand integrating both sides:

281


Example. Which of these equations are separable?

1. y′ =x√y

2. y′ = x + y

3. y′ = yex+y

4. y′ = ln(xy)

5. y′ = ln(xy)

6. y′ =xy + y

2x − 3xy7. y′ = xy − 2x + y − 2

282


Example. d2y

dx2 = −4y1. Show that an equation of this form describes the motion of a spring.

2. Find as many solutions as possible for this equation.

283


Example. y′ = x + y

1. This equation is harder to solve or guess solutions for, but we can get approximatesolutions by plotting the “slope field”.

x y y′ (note: y′ = x + y)

284


Slope field for y′ = x + y

2. Sketch some curves whose tangent lines fall on this slope field.

3. Sketch an approximate solution to the differential equation that satisfies the initialcondition y(−1) = 1.

285


Example. For each situation, set up a differential equation. If you have extra time atthe end, you can solve the equations.

1. The rate of cooling of an object is proportional to the temperature difference be-tween the object and its surroundings. Write a differential equation to describe thetemperature of a cup of coffee that starts out at 90◦ C and is in a 20◦ room.

2. A population is growing at a rate proportional to the population size .

3. The logistic population model assumes that there is a maximum carrying capacityof M and that the rate of change of the population is proportional to the productof the population and the fraction of the carrying capacity that is left.

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§6.1 - Area Between Curves §6.2 - Volumes§6.4 - Work§6.5 - Average Value of a Function§7.1 - Integration by Parts§7.2 - Integrating Trig Functions§7.3 - Trig Substitutions§7.4 - Integrals of Rational Functions§7.5 Integration StrategiesPhilosophy about Integration§7.8 -Improper Integrals§11.1 - Sequences§11.2 - Series§11.3 - The Integral Test§11.4 - Comparison Tests for SeriesSection 11.5 - Alternating Series§11.6 - Ratio and Root Tests§11.7 - Strategy for Convergence Tests for Series§11.8 - Power Series§11.9 - Approximating functions with power series§11.10 - Taylor Series§11.10 and §11.11 - Taylor Series Theory and Remainders§8.1 - Arclength§10.1 - Parametric Equations§10.2 Calculus using Parametric Equations§10.3 Polar Coordinates§10.4 Area in polar coordinates§4.4 - L'Hospital's Rule§9.1, 9.2, 9.3 - Differential Equations

Math 232 Calculus 2 - Spring 2016 - Linda...

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