Deck 7: Additional Topics in Integration

Question 1

Find the volume of the solid of revolution obtained by revolving the region under the curve
​ ​
from to about the -axis.

Answer 1

Question 2

By computing the volume of the solid of revolution obtained by revolving the region under the semicircle
​ ​
from to about the -axis, show that the volume of a sphere of radius is cubic units.

Answer 2

cu units

Question 3

Find the volume of the solid of revolution obtained by revolving the region under the curve
​ ​
from to about the -axis.

Answer 3

Question 4

Find the volume of the prolate spheroid (a solid of revolution in the shape of a football, i.e., elongated vertically) obtained by revolving the region under the graph of the function
​ ​
from to about the -axis.

Question 5

Find the volume of the solid of revolution obtained by revolving the region under the curve
​ ​
from to about the -axis.

Question 6

Find the volume of the solid of revolution obtained by revolving the region bounded above by the curve
​ ​
and below by the curve
​ ​
from to about the -axis.

Question 7

Find the volume of the solid of revolution obtained by revolving the region bounded by the graphs of the given functions about the x-axis.
​ and

Question 8

Find the volume of the solid of revolution obtained by revolving the region bounded by the graphs of the given functions about the x-axis.
​ , , and ​

Question 9

Find the volume of the solid of revolution obtained by revolving the region bounded above by the curve
​ ​
and below by the curve
​ ​
from to about the -axis.

Question 10

Find the volume of the solid of revolution obtained by revolving the region bounded above by the curve
​ ​
and below by the curve
​ ​
from to about the -axis.

Question 11

Find the volume of the solid of revolution obtained by revolving the region under the curve
​ ​
from to about the -axis.

Question 12

Find the volume of the solid of revolution obtained by revolving the region bounded above by the curve
​ ​
and below by the curve
​ ​
from to about the -axis.

Question 13

Find the volume of the solid of revolution obtained by revolving the region under the curve
​ ​
from to about the -axis.

Question 14

Find the volume of the solid of revolution obtained by revolving the region bounded by the graphs of the given functions about the x-axis.
​ and

Question 15

Find the volume of the solid of revolution obtained by revolving the region under the curve
​ ​
from to about the -axis.

Question 16

Find the volume of the solid of revolution obtained by revolving the region bounded by the graphs of the given functions about the x-axis.
​ and

Question 17

Find the volume of the solid of revolution obtained by revolving the region bounded above by the curve
​ ​
and below by the curve
​ ​
from to about the -axis.

Question 18

Find the volume of the solid of revolution obtained by revolving the region under the curve
​ ​
from to about the -axis.

Question 19

Find the volume of the solid of revolution obtained by revolving the region bounded by the graphs of the given functions about the x-axis.
​ and

Question 20

Find the volume of the solid of revolution obtained by revolving the region under the curve
​ ​
from to about the -axis.

Question 21

Find the area of the region under the given curve over the indicated interval. ​ ​

A)
B) ​
C) ​
D) ​

Question 22

Evaluate the following improper integral whenever it is convergent. ​ ​

A)
B)
C)
D)
E) the integral is divergent

Question 23

Evaluate the improper integral if it is convergent. ​ ​

A) 4
B) -3
C) 2
D) 3
E) divergent

Question 24

Find the area of the region under the given curve over the indicated interval. ​ ​

A)
B)
C)
D) ​
E) ​

Question 25

Evaluate the following improper integral whenever it is convergent. ​ ​

A)
B) ​
C) ​
D) ​
E) the integral is divergent

Question 26

Evaluate the following improper integral whenever it is convergent. ​ ​

A) 2
B) -8
C) 4
D) 8
E) the integral is divergent

Question 27

Evaluate the following improper integral whenever it is convergent. ​ ​

A) -6
B) 11
C) 3
D) 7
E) the integral is divergent

Question 28

Find the area of the region under the given curve over the indicated interval. ​ ​

A) 2 sq units
B) 5 sq units
C) 6 sq units
D) 7 sq units

Question 29

Evaluate the following improper integral whenever it is convergent. ​ ​

A) 6
B) 4
C) 2
D) 10
E) the integral is divergent

Question 30

If exists, then exists and .

Question 31

Evaluate the following improper integral whenever it is convergent. ​ ​

A)
B)
C)
D) ​
E) the integral is divergent

Question 32

Evaluate the improper integral whenever it is convergent. ​ ​

A)
B) ​
C) ​
D) ​
E) divergent

Question 33

Find the area of the region under the given curve over the indicated interval. ​ ​

A)
B)
C) ​
D) ​
E) ​

Question 34

Find the area of the region bounded by the x-axis and the graph of the function ​ ​

A)
B)
C) ​
D) ​

Question 35

Evaluate the following improper integral whenever it is convergent. ​ ​

A)
B) ​
C) ​
D) ​
E) the integral is divergent

Question 36

Evaluate the following improper integral whenever it is convergent. ​ ​

A) 8
B) 0
C) -6
D) -10
E) the integral is divergent

Question 37

Evaluate the following improper integral whenever it is convergent. ​ ​

A) 4
B) 10
C) 2
D) 5
E) the integral is divergent

Question 38

Find the area of the region bounded by the x-axis and the graph of the function. ​ ​

A) 1 sq units
B) 5 sq units
C) 4 sq units
D) 3 sq units
E) 2 sq units

Question 39

Find the area of the region under the curve over the indicated interval. ​ ​

A) 2 sq units
B) 3 sq units
C) 4 sq units
D) 8 sq units
E) 1 sq units

Question 40

Evaluate the following improper integral whenever it is convergent. ​ ​

A)
B)
C) 1
D) 0
E) the integral is divergent

Question 41

Becky Wilkinson wants to establish a trust fund that will provide her children and heirs with a perpetual annuity in the amount of ​ ​
Thousand dollars per year beginning next year. If the trust fund will earn an interest rate of 10% per year compounded continuously, find the amount that she must place in the trust fund now.
Hint: Use the formula

A) $350,000
B) $290,000
C) $380,000
D) $370,000
E) $400,000

Question 42

Evaluate the improper integral if it is convergent.
​

Question 43

Find the area of the region bounded by the x-axis and the graph of the function.
​ ​
__________ sq units

Question 44

Find the area of the region under the given curve over the indicated interval.
​

Question 45

Find the area of the region under the curve over the indicated interval.
​ ​
__________ sq units

Question 46

Evaluate the following improper integral whenever it is convergent.
​

Question 47

Becky Wilkinson wants to establish a trust fund that will provide her children and heirs with a perpetual annuity in the amount of
​ ​
thousand dollars per year beginning next year. If the trust fund will earn an interest rate of 10% per year compounded continuously, find the amount that she must place in the trust fund now.
​

Question 48

Find the area of the region under the given curve over the indicated interval.
​

Question 49

Find the area of the region under the given curve over the indicated interval.
​

Question 50

Find the area of the region under the given curve over the indicated interval.
​

Question 51

Find the area of the region bounded by the x-axis and the graph of the function
​

Question 52

Evaluate the following improper integral whenever it is convergent.
​

Question 53

A university alumni group wishes to provide an annual scholarship in the amount of $1,100 beginning next year. If the scholarship fund will earn an interest rate of 5% / year compounded continuously, find the amount of the endowment the alumni are required to make now. ​

A) $22,559
B) $21,118
C) $22,000
D) $21,556

Question 54

Evaluate the improper integral whenever it is convergent. If not, enter divergent.
​

Question 55

Evaluate the following improper integral whenever it is convergent.
​

Question 56

A university alumni group wishes to provide an annual scholarship in the amount of $1,900 beginning next year. If the scholarship fund will earn an interest rate of 4% / year compounded continuously, find the amount of the endowment the alumni are required to make now.
​
$__________

Question 57

Evaluate the following improper integral whenever it is convergent.
​ ​
Enter divergent if the integral is divergent.

Question 58

The present value of a perpetual stream of income that flows continually at the rate of P(t) dollars per year is given by the formula ​ ​
Where r is the interest rate compounded continuously. Using this formula, find the present value of a perpetual net income stream that is generated at the rate of dollars per year.
Hint:

A)
B)
C)
D)
E)
​

Question 59

Evaluate the following improper integral whenever it is convergent.
​

Question 60

Evaluate the following improper integral whenever it is convergent.
​

Question 61

The present value of a perpetual stream of income that flows continually at the rate of P(t) dollars per year is given by the formula
​ ​
where r is the interest rate compounded continuously. Using this formula, find the present value of a perpetual net income stream that is generated at the rate of
​ ​
dollars per year.
​

Question 62

If exists, then exists and .

Question 63

Evaluate the following improper integral whenever it is convergent.
​ ​
Enter divergent if the integral is divergent.

Question 64

Use the trapezoidal rule and Simpson's rule to approximate the value of the definite integral. ​ ​

A)
B) ​
C) ​
D) ​

Question 65

Evaluate the following improper integral whenever it is convergent.
​ Enter divergent if the integral is divergent.

Question 66

Find a bound on the error in approximating the given definite integral using (a) the trapezoidal rule and (b) Simpson's rule with n intervals. ​ ​

A)
B) ​
C) ​
D) ​

Question 67

Find a bound on the error in approximating the definite integral using the trapezoidal rule with 8 intervals. ​ ​

A) 0.1773108
B) 0.1110032
C) 0.0629052
D) 0.0625000
E) 0.0403685

Question 68

Evaluate the following improper integral whenever it is convergent.
​ ​
Enter divergent if the integral is divergent.

Question 69

Use the trapezoidal rule and Simpson's rule to approximate the value of the definite integral. Find the exact value of the integral. ​ ​

A)
B) ​
C) ​
D) ​

Question 70

In a submerged trial run of an attack submarine, a reading of the submarine's velocity was made every quarter hour, as shown in the accompanying table. Use the trapezoidal rule to estimate the distance traveled by the submarine during the 2-hour period. Please round the answer to the nearest hundredth if necessary. Time, t (hr)
0 Velocity, V(t) (mph)
19)4
23
33)8
39)4
37)2
35)1
27)8
26)2
3)8
​

A) 52.89 mi
B) 54.15 mi
C) 58.52 mi
D) 45.47 mi
E) 47.87 mi

Question 71

Evaluate the following improper integral whenever it is convergent.
​ ​
Enter divergent if the integral is divergent.

Question 72

Use the trapezoidal rule and Simpson's rule to approximate the value of the definite integral. ​ ​

A) By trapezoidal rule 1.1325, by Simpson's rule 0.6554
B) By trapezoidal rule 1.9565, by Simpson's rule 0.6555
C) By trapezoidal rule 0.6572, by Simpson's rule 0.6551
D) By trapezoidal rule 1.6290, by Simpson's rule 1.6108
E) By trapezoidal rule 1.5807, by Simpson's rule 0.6533

Question 73

Use the trapezoidal rule and Simpson's rule to approximate the value of the definite integral. Please give the answer to four decimal places. ​ ​

A) By trapezoidal rule 1.0271, by Simpson's rule 1.1893
B) By trapezoidal rule 0.9083, by Simpson's rule 0.7109
C) By trapezoidal rule 1.1902, by Simpson's rule 1.1677
D) By trapezoidal rule 0.8126, by Simpson's rule 0.5496
E) By trapezoidal rule 1.4060, by Simpson's rule 0.7978

Question 74

Use the trapezoidal rule and Simpson's rule to approximate the value of the definite integral. ​ ​

A) By trapezoidal rule 1.2140, by Simpson's rule 1.1609
B) By trapezoidal rule 1.1705, by Simpson's rule 1.1520
C) By trapezoidal rule 1.1000, by Simpson's rule 1.1660
D) By trapezoidal rule 1.3620, by Simpson's rule 1.1306
E) By trapezoidal rule 0.9613, by Simpson's rule 0.9503

Question 75

Use the trapezoidal rule and Simpson's rule to approximate the value of the definite integral. Find the exact value of the integral. ​ ​

A)
B) ​
C) ​
D) ​

Question 76

Use the trapezoidal rule and Simpson's rule to approximate the value of the definite integral. Find the exact value of the integral. ​ ​

A)
B) ​
C) ​
D) ​

Question 77

Evaluate the following improper integral whenever it is convergent.
​ ​
Enter divergent if the integral is divergent.

Question 78

Use the trapezoidal rule and Simpson's rule to approximate the value of the definite integral. ​ ​

A)
B)
C)
D)

Question 79

Use the trapezoidal rule and Simpson's rule to approximate the value of the definite integral. Find the exact value of the integral. ​ ​

A)
B) ​
C) ​
D) ​

Question 80

Find a bound on the error in approximating the given definite integral using (a) the trapezoidal rule and (b) Simpson's rule with n intervals. ​ ​

A)
B) ​
C) ​
D) ​