Deck 7: Techniques of Integration

Question 1

Determine whether the improper integral converges or diverges, and if it converges, find its value.

A)
B)Diverges
C)
D)

Answer 1

Question 2

Evaluate the integral or show that it is divergent.

A)
B)
C)
D)
E)

Answer 2

Question 3

Determine whether the improper integral converges or diverges, and if it converges, find its value.

A)0
B)4
C)Diverges
D)2

Answer 3

Diverges

Question 4

Determine whether the improper integral converges or diverges, and if it converges, find its value.

A)3
B)Diverges
C)6
D)0

Question 5

Find the area bounded by the curves and between and .

A)
B)
C)
D)
E)

Question 6

Determine whether the integral converges or diverges. If it converges, find its value.

Question 7

Evaluate the integral.

A)
B)
C)
D)
E)

Question 8

Evaluate the integral or show that it is divergent.

Question 9

Evaluate the integral.

A)
B)
C)
D)
E)

Question 10

For what values of K is the following integral improper?

Question 11

A manufacturer of light bulbs wants to produce bulbs that last about hours but, of course, some bulbs burn out faster than others. Let be the fraction of the company's bulbs that burn out before t hours. lies between 0 and 1. Let . What is the value of ?

A)
B)
C)
D)
E)

Question 12

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places.

A)0.2326
B)0.1088
C)0.1163
D)0.2029

Question 13

Evaluate the integral.

Question 14

Determine whether the improper integral converges or diverges, and if it converges, find its value.

Question 15

The region is represented below. Find the area of this region to two decimal places.

A) 15.87
B) 17.89
C)20.28
D)
E)

Question 16

Evaluate the integral if it is convergent.

Question 17

Determine whether the improper integral converges or diverges, and if it converges, find its value.

A)Diverges
B)
C)
D)

Question 18

Determine whether the improper integral converges or diverges, and if it converges, find its value.

A)
B)Diverges
C)0
D)

Question 19

Let a and b be real numbers. What integral must appear in place of the question mark "?" to make the following statement true?

A)
B)
C)
D)
E)

Question 20

Determine whether the improper integral converges or diverges, and if it converges, find its value.

Question 21

Estimate the area of the shaded region by using the Trapezoidal Rule with . Round the answer to the nearest tenth.

Question 22

Use a table of integrals to evaluate the integral.

A)
B)
C)
D)

Question 23

A body moves along a coordinate line in such a way that its velocity at any time t, where , is given by .
Find its position function if it is initially located at the origin.

Question 24

The region under the curve , is rotated about the x-axis. Find the volume of the resulting solid.

A)
B)
C)
D)
E)

Question 25

Find a bound on the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule with n subintervals.

Question 26

Find a bound on the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule with subintervals.

Question 27

Find a bound on the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule with n subintervals.

Question 28

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral to four decimal places. Compare your results with the exact value.

Question 29

Use the Trapezoidal Rule to approximate for . Round the result to four decimal places.

Question 30

Use the Table of Integrals to evaluate the integral.

A)
B)
C)
D)
E)

Question 31

Eight milligrams of a dye are injected into a vein leading the an individual's heart. The concentration of dye in the aorta (in milligrams per liter) measured at 2-sec intervals is shown in the accompanying table. Use Simpson's Rule with and the formula to estimate the person's cardiac output, where D is the quantity of dye injected in milligrams, is the concentration of the dye in the aorta, and R is measured in liters per minute. Round to one decimal place.

Question 32

Use the Trapezoidal Rule to approximate the integral with answers rounded to four decimal places.

A)0.8528
B)0.9842
C)0.4921
D)0.4695

Question 33

Use the Midpoint Rule to approximate the given integral with the specified value of n. Compare your result to the actual value. Find the error in the approximation.

A)
B)
C)1.00008
D)0.00008
E)

Question 34

Use the Table of Integrals to evaluate the integral.

A)
B)
C)
D)
E)

Question 35

Use a table of integrals to evaluate the integral.

A)
B)
C)
D)

Question 36

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places.

A)4.0689
B)3.6993
C)3.3296
D)2.9599

Question 37

Use a table of integrals to evaluate the integral.

A)
B)
C)
D)

Question 38

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places.

A)2.6098
B)2.9504
C)2.0076
D)2.2955

Question 39

Use Simpson's Rule to approximate the integral with answers rounded to four decimal places.

A)3.0265
B)2.2028
C)2.4774
D)2.7519

Question 40

Find a bound on the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule with n subintervals.

Question 41

Evaluate the integral.

Question 42

Find the integral.

A)
B)
C)
D)

Question 43

Find the integral.

A)
B)
C)
D)

Question 44

Evaluate the integral.

Question 45

Find the integral.

A)
B)
C)
D)

Question 46

Evaluate the integral.

Question 47

Find the integral.

A)
B)
C)
D)

Question 48

Evaluate the integral.

Question 49

Use the Table of Integrals to evaluate the integral to three decimal places.

Question 50

Write the form of the partial fraction decomposition of the rational expression. Do not find the numerical values of the constants.

A)
B)
C)
D)

Question 51

Evaluate the integral.

Question 52

Use a table of integrals to evaluate the integral.

A)
B)
C)
D)

Question 53

Use the Table of Integrals to evaluate the integral.

Question 54

Evaluate the integral.

A)
B)
C)
D)
E)

Question 55

Evaluate the integral.

Question 56

Evaluate the integral.

A)
B)
C)
D)
E)

Question 57

Use long division to evaluate the integral.

A)
B)
C)
D)
E)

Question 58

Use a table of integrals to evaluate the integral.

A)
B)
C)
D)

Question 59

Evaluate the integral.

A)
B)
C)
D)
E)

Question 60

Evaluate the integral.

Question 61

Find the integral using an appropriate trigonometric substitution.

A)
B)
C)
D)

Question 62

Use long division to evaluate the integral. The choices are rounded to 3 decimal places.

A)
B)
C)
D)
E)

Question 63

Evaluate the integral.

A)
B)
C)
D)
E)

Question 64

The region under the graph of on the interval [1, 2] is revolved about the x-axis. Find the volume of the resulting solid.

Question 65

Eight milligrams of a dye are injected into a vein leading the an individual's heart. The concentration of dye in the aorta (in milligrams per liter) measured at 2-sec intervals is shown in the accompanying table. Use Simpson's Rule with and the formula to estimate the person's cardiac output, where D is the quantity of dye injected in milligrams, is the concentration of the dye in the aorta, and R is measured in liters per minute. Round your answer to one decimal place.

Question 66

Evaluate the integral using the indicated trigonometric substitution.

Question 67

Find the integral using an appropriate trigonometric substitution.

A)
B)
C)
D)

Question 68

Evaluate the integral using an appropriate trigonometric substitution.

A)
B)
C)
D)

Question 69

Evaluate the integral.

Question 70

Find the integral using an appropriate trigonometric substitution.

A)
B)
C)
D)

Question 71

Evaluate the integral using the indicated trigonometric substitution.

A)
B)
C)
D)
E)

Question 72

A corporation is building a complex of homes, offices, stores, schools, and churches in a rural community. As a result of this development, the planners have estimated that the community's population (in thousands) t years from now will be given by .
What will the average population of the community be over the next 10 years?

Question 73

Find the integral.

Question 74

Make a substitution to express the integrand as a rational function and then evaluate the integral. Round the answer to four decimal places.

Question 75

Find the integral.

Question 76

Evaluate the integral.

A)
B)
C)
D)
E)

Question 77

Find the integral using an appropriate trigonometric substitution.

A)
B)
C)
D)

Question 78

Find the integral.

Question 79

Find the integral.

A)
B)
C)
D)

Question 80

Find the volume of the resulting solid if the region under the curve from to is rotated about the x-axis. Round your answer to four decimal places.