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p265 Section 4.3: Reimann Sums and Definite Integrals

Power Point Notes:

Example 2: Evaluating a Definite Integral

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Example 3: Areas of Common Geometric FiguresSketch the region corresponding to each definite integral. Then calculate each 

integral using a geometric formula

a. What is this shape?  Rectangle

What is the formula to find the area? A = lw

Find the area of this shape:  A = 2*4 = 8

b.

What is this shape?  Trapezoid

What is the formula to find the area? A = (1/2)h(b1 + b2)

Find the area of this shape:  A = (1/2)(3)(2 + 5) = 21/2 = 10.5

b1

b2

h

c.

What is this shape?  Semicircle  r = 2

What is the formula to find the area? A = (1/2)πr2

Find the area of this shape:  A = (1/2)π(2)2 = 2π

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Example 4: Evaluating Definite Integrals

a.

b. This is the same at the integral in Example 3b except the upper and lower limit have been interchanged. Remember the integral in Example 3b was equal to 21/2

Example 5: Using the Additive Interval Property

What are the shapes? Triangles

Can you calculate the area of each shape?

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Example 6: Evaluation of a Definite IntegralEvaluate the integral using each of the following values

Evaluate the definite integral

#3.

#5.

#7.

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Express the limit as a definite integral on the interval [a, b], where ci is any point in the ith subinterval

Limit Interval

#9. [1, 5]

#11. [0, 3]

Now look at #13  21 in your book (p272) and set up a definite integral that would yield the area of the region on each graph

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Sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral (a > 0, r > 0)

#23.Find the Area of a rectangle (A = lw)

A = 3*4 = 12

#25. Find the Area of a Triangle (A = (1/2)bh)

A = (1/2)(4)(4) = 8

#27.

Find the Area of a TrapezoidA = (1/2)h(b1 + b2)

A = (1/2)2(5 + 9) = (1/2)(2)(14) = 14

h

b1b2

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#29.

Find the Area of the TriangleA = (1/2)bh

A = (1/2)(2)(1) = 1

#31. Find the Area of the Semicircle  (r = 3)

A = (1/2)πr2

A = (1/2)π(3)2 = 9π/2

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#41.  Use the given information to find each of the following:

(a)

(b)

(c)

(d)

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#45.

(a)

(b)

(c)

(d)

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#45.  The graph of f consists of line segments and a semicircle, as shown in the figure. Evaluate each definite integral by using geometric formulas.

(4, 2)

(4, 1)

(a) Area of a Quarter Circle (below xaxis  negative)

(b) Area of Triangle

(c) Area of Triangle and Semicircle (below xaxis  negative)

(d) Area below xaxis (from [4, 2]) + Area above xaxis (from [2, 6])Area from part (b) + part (c)

(e)

(f)

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HW p272 #4  10 (even), 24, 28, 42, 44

Attachments

Section 4.3  Reimann Sums and Definite Integrals.pptx

Riemann Sum

When we find the area under a curve by adding rectangles, the answer is called a Rieman sum.

subinterval

partition

The width of a rectangle is called a subinterval.

The entire interval is called the partition.

Subintervals do not all have to be the same size.

subinterval

partition

If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by .

As gets smaller, the approximation for the area gets better.

if P is a partition

of the interval

is called the definite integral of

over .

If we use subintervals of equal length, then the length of a subinterval is:

The definite integral is then given by:

Leibnitz introduced a simpler notation for the definite integral:

Note that the very small change in x becomes dx.

Limit of Riemann Sum = Definite Integral

Integration

Symbol

lower limit of integration

upper limit of integration

integrand

variable of integration

(dummy variable)

It is called a dummy variable because the answer does not depend on the variable chosen.

The Definite Integral

Existence of Definite Integrals

All continuous functions are integrable.

Example Using the Notation

Area Under a Curve (as a Definite Integral)

Note: A definite integral can be positive, negative or zero, but for a definite integral to be interpreted as an area the function MUST be continuous and nonnegative on [a, b].

Area

Integrals on a Calculator

Example Using NINT

Properties of Definite Integrals

Order of Integration

Zero

Constant Multiple

Sum and Difference

Additivity

If f is integrable and nonnegative on the closed interval [a, b], then

If f and g are integrable on the closed interval [a, b] and for every x in [a, b], then

Preservation of Inequality

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