Antiderivatives(7.4, 8.2, 10.1)

JMerrill, 2009

Review Info - Antiderivatives

General solutions:

y f(x)dx F(x) C Integrand

Variable of Integration

Constant of Integration

Review

Rewriting & Integrating – general solution

Original Rewrite Integrate Simplify

31

dxx 3x dx

2xC

2

2

1C

2x

x dx12x dx

32x

C32

322

x C3

Particular Solutions

To find a particular solution, you must have an initial condition

Ex: Find the particular solution of that satisfies the condition F(1) = 0

2

1F'(x)

x

2

1F'(x) dx

x

2x dx1x 1

C C1 x

1F(1) C

1

1F(x) C

x

0 1 C

1 C

1F(x) 1

x

Indefinite & Definite Integrals

Indefinite Integrals have the form:

Definite integrals have the form:

f (x)dxb

a

f (x)dx

7.4 The Fundamental Theorem of Calculus This theorem represents the relationship

between antiderivatives and the definite integral

Here’s How the Theorem Works

First find the antiderivative, then find the definite integral

2

3

1

4x dx4

3 44x4x dx x

4

24

1x 4 42 1 15

Properties of Definite Integrals

The chart on P. 466:

Example – Sum/Difference

Find 5

2

2

6x 3x 5 dx 5 5 5

2

2 2 2

6 x dx 3 xdx 5 dx

32 3

22

x6 x 6 2x

3

x 33 xdx 3 x

2 2

5 dx 5 x 5x

55 53 222 2

32x x 5x

2

3 3 2 232 5 2 5 2 5 5 2

2

63234 15

24352

Less Confusing Notation?

Evaluate 2

2

0

2x 3x 2 dx 23 2

0

2x 3x2x

3 2

166 4 0 0 0

3

103

Substitution - Review

Evaluate

Let u = 3x – 1; du = 3dx

43 3x 1 dx

4u du

455u

33

x 1 3dxx

C1

55C

Substitution & The Definite Integral

Evaluate

Let u = 25 – x2; du = -2xdx

5

2

0

x 25 x dx

5

2

0

5 5 12

0 0

125 x 2xdx

2

1 1udu u du

2 2

32u

C3

53

2 2

0

25 x

3

32250 1

335

32

321 u

C322

Area

Find the area bounded by the curve of f(x) = (x2 – 4), the x-axis, and the vertical lines x = 0, x = 2

2

2

0

x 4 dx23

0

x4x

3

88 0

3163

0

The answer is negative because the area is below the x-axis. Since area must be positive just take the absolute value.

163

Finding Area

Area – Last Example

Find the area between the x-axis and the graph of f(x) = x2 – 4 from x = 0 to x = 4.

2 4

2 2

0 2

x 4 dx x 4 dx 2 4

3 3

0 2

1 1x 4x x 4x

3 3

8 64 88 0 0 16 8

3 3 3

16

8.2 Volume & Average Value

We have used integrals to find the area of regions. If we rotate that region around the x-axis, the resulting figure is called a solid of revolution.

Volume of a Solid of Revolution

Volume Example

Find the volume of the solid of revolution formed by rotating about the x-axis the region bounded by y = x + 1, y = 0, x = 1, and x = 4.

Volume Example

4

2

1

V x 1 dx

43

1

x 1

3

3 3 1175 2

3 339

Volume Problem

Find the volume of the solid of revolution formed by rotating about the x-axis the area bounded by f(x) = 4 – x2 and the x-axis.

Volume Con’t

2

22

2

V 4 x dx

2

2 4

2

16 8x x dx

23 5

2

8x x16x

3 5

51264 32 64 3232 32

3 5 5 153

Average Value

Average Price

A stock analyst plots the price per share of a certain common stock as a function of time and finds that it can be approximated by the function

S(t)=25 - 5e-.01t

where t is the time (in years) since the stock was purchased. Find the average price of the stock over the first 6 years.

Avg Price - Solution

We are looking for the average over the first 6 years, so a = 0 and b = 6.

6 .01

0

125 5

6 0

te dt

The average price of the stock is about $20.15

6.01

0

1 525

6 .01

tt e

.061150 500 500

620.147

e

10.1 Differential Equations

A differential equation is one that involves an unknown function y = f(x) and a finite number of its derivatives. Solving the differential equation is used for forecasting interest rates.

A solution of an equation is a number (usually).

A solution of a differential equation is a function.

Differential Equations

Population Example

The population, P, of a flock of birds, is growing exponentially so that , where x is time in years.

Find P in terms of x if there were 20 birds in the flock initially.

0.05xdP20e

dx

Note: Notice the denominator has the same variable as the right side of the equation.

Population Cont

Take the antiderivative of each side:

This is an initial value problem. At time 0, we had 20 birds.

0.05xP 20e dx0.05xdP

20edx

0.05x 0.05x20e C 400e C

0.05

0 20 400e C

380 C

0.05xP 400e 380

One More Initial Value Problem

Find the particular solution of when y = 2, x = -1

dy2x 5

dx

dy2x 5

dx

dy2x 5dx

dx

222x

y 5x C x 5x C2

22 ( 1) 5( 1) C

6 C

2y x 5x 6

Note: Notice the denominator has the same variable as the right side of the equation.

Separation of Variables

Not all differential equations can be solved this easily.

If interest is compounded continuously then the money grows at a rate proportional to the amount of money present and would be modeled by dA

kAdt

Note: Notice the denominator does not have the same variable as the right side of the equation.

Separation of Variables

In general terms think of

This of dy/dx as the fraction dy over dx (which is totally incorrect, but it works!)

In this case, we have to separate the variables

dy f(x)dx g(y)

g(y)dy f(x)dx

G(y) F(x) C

(Get all the y’s on one side and all the x’s on the other)

Example

Find the general solution of

Multiply both sides by dx to get

2dyy x

dx

2y dy x dx

2y dy x dx 2 3y x

C2 3

2 32y x 2C

3

Lab 4 – Due Next Time on Exam Day

1. #34, P471 9. #25, P523 2. #59, P440 10. #35, P523 3. #22, P471 11. #3, P629 4. #45, P439 12. #7, P629 5. #11, P439 13. #19, P630 6. #13, P471 14. #27, P630 7. #27, P439 15. #43, P472 8. #17, P522