Stewart – Calculus ET 8e Chapter 8 Form C
Answer Key
Stewart – Calculus ET 8e Chapter 8 Form D
Select the correct answer for each question.
____ 1. Find the length of the curve.
a.
b.
c.
d.
e. None of these
____ 2. Find the area of the surface obtained by rotating the circle about the line .
a.
b.
c.
d.
e.
____ 3. A trough is filled with a liquid of density 855 kg/ . The ends of the trough are equilateral triangles with
sides m long and vertex at the bottom. Find the hydrostatic force on one end of the trough.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 8 Form D
____ 4. Find the arc length of the graph of the given equation on the specified interval.
y =
2
3
(+ 1) , [2, 3]
a.
41
3
b.
41
2
c.
41
4
d.
41
6
____ 5. Set up, but do not evaluate, an integral for the length of the curve.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 8 Form D
____ 6. Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve
about the x-axis on the interval .
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 8 Form D
____ 7. Write an integral giving the area of the surface obtained by revolving the curve about the x-axis. (Do not
evaluate the integral.)
y = on [1, 5]
a.
b. 4
c.
4
d.
____ 8. Find the centroid of the region bounded by the graphs of the given equations.
a.
b.
c.
d.
Stewart – Calculus ET 8e Chapter 8 Form D
____ 9. The masses are located at the points . Find the moments and and the center of mass of the
system.
;
a.
b.
c.
d.
e.
____ 10. The marginal revenue from producing x units of a certain product is
(in dollars per unit).
Find the increase in revenue if the production level is raised from 1,100 units to units.
a. $37,974,583
b. $26,866,667
c. $17,765,250
d. $
e. $51,367,000
Stewart – Calculus ET 8e Chapter 8 Form D
____ 11. The marginal cost function is defined to be the derivative of the cost function. If the marginal
cost of manufacturing x units of a product is
(measured in dollars per unit) and the fixed start-up cost is , use the Total Change
Theorem to find the cost of producing the first 5,000 units.
a. $354,445,000
b. $354,345,000
c. $354,845,000
d. $354,545,000
e. $
____ 12. A vertical plate is submerged in water (the surface of the water coincides with the x-axis). Find the force
exerted by the water on the plate. (The weight density of water is 62.4 lb/ft3.)
(f
t
)
12345 x
1
–1
–2
–3
–4
y
(ft)
a. 624 lb
b. 62 lb
c. 125 lb
d. 312 lb
____ 13. Find the length of the curve.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 8 Form D
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____ 14. If f (x) is the probability density function for the blood cholesterol level of men over the age of 40,
where x is measured in milligrams per deciliter, express as an integral the probability that the
cholesterol level of such a man lies between 195 and .
a.
b.
c.
d.
e.
____ 15. Dye dilution is a method of measuring cardiac output. If A mg of dye is used and c(t) is the
concentration of the dye at time t, then the cardiac output over the time interval [0, T ] is given by
Find the cardiac output over the time interval [0, 15] if the dye dilution method is used with mg
of dye and the dye concentration, in mg/L, is modeled by
,
where t is measured in seconds.
a. 0.031
b. 0.043
c. 0.029
d. 0.049
e.
f. 0.022
Stewart – Calculus ET 8e Chapter 8 Form D
____ 16. Find the centroid of the region bounded by the given curves.
a.
b.
28
75
,
92
105
c.
d.
e. None of these
____ 17. Find the area of the surface obtained by revolving the given curve about the x-axis.
on
a.
b.
c.
d.
____ 18. Use the Theorem of Pappus to find the volume of the solid obtained by revolving the region bounded by the
graphs of and about the y-axis.
a.
128
b.
512
3
c.
128
3
d.
32
Stewart – Calculus ET 8e Chapter 8 Form D
____ 19. According to the National Health Survey, the heights of adult males in the United States are
(normally distributed with mean) 73 inches, and standard deviation of 2.8 inches. What is the
probability that an adult male chosen at random is between inches and inches tall?
a. 0.759
b.
c. 0.2812
d. 0.6512
e. 0.7154
f. 0.5528
____ 20. The standard deviation for a random variable with probability density function f and mean µ is
defined
.
Find the standard deviation for an exponential density function with mean .
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 8 Form D
Answer Key
Stewart – Calculus ET 8e Chapter 8 Form E
____ 1. Find the length of the curve. Select the correct answer.
a.
b. 25.05
c. 13.05
d. 36.05
e. None of these
____ 2. Use Simpson’s Rule with n = 6 to estimate the length of the curve , .
Round your answer to six decimal places. Select the correct answer.
a. 6.987208
b. 6.947368
c. 6.972089
d. 6.947582
e. None of these
3. Find the arc length of the graph of the given equation on the specified interval.
y =
2
3
(+ 1) , [2, 5]
____ 4. Use differentials to approximate the arc length of the graph of the equation from P to Q. Round answer to
four decimal places. Select the correct answer.
P (3, 12), Q (3.2, 13.24)
a. 6.0828
b. 3.6497
c. 1.2166
d. 2.4331
5. Find the arc length function for the curve with starting point .
6. Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve
about the x-axis on the interval .
7. Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve about the
given axis.
y-axis
Stewart – Calculus ET 8e Chapter 8 Form E
8. Find the area of the surface obtained by rotating the curve about the x-axis.
____ 9. Find the area of the surface obtained by rotating the circle about the line .
Select the correct answer.
a.
b.
c.
d.
e.
____ 10. If the infinite curve , is rotated about the x-axis , find the area of the resulting surface.
Select the correct answer.
a.
b.
c.
d.
e.
____ 11. A trough has vertical ends that are equilateral triangles with sides of length 7 ft. If the trough is filled with
water to a depth of 4 ft, find the force exerted by the water on one end of the trough. Round to one decimal
place. (The weight density of water is 62.4 lb/ft3.) Select the correct answer.
a. 384.3 lb
b. 1,337.7 lb
c. 768.6 lb
d. 2,675.4 lb
Stewart – Calculus ET 8e Chapter 8 Form E
12. Find the centroid of the region bounded by the graphs of the given equations.
13. Find the centroid of the region bounded by the given curves.
14. Find the centroid of the region bounded by the graphs of the given equations.
____ 15. The marginal cost function is defined to be the derivative of the cost function. If the marginal
cost of manufacturing x units of a product is
(measured in dollars per unit) and the fixed start-up cost is , use the Total Change
Theorem to find the cost of producing the first 5,000 units. Select the correct answer.
a. $354,645,000
b. $354,845,000
c. $
d. $354,545,000
e. $354,745,000
16. Dye dilution is a method of measuring cardiac output. If A mg of dye is used and c(t) is the
concentration of the dye at time t, then the cardiac output over the time interval [0, T ] is given by
Find the cardiac output over the time interval [0, 15] if the dye dilution method is used with mg
of dye and the dye concentration, in mg/L, is modeled by
,
where t is measured in seconds.
Stewart – Calculus ET 8e Chapter 8 Form E
____ 17. The marginal revenue from producing x units of a certain product is
(in dollars per unit).
Find the increase in revenue if the production level is raised from 1,100 units to units.
Select the correct answer.
a. $37,974,583
b. $51,367,000
c. $26,866,667
d. $17,765,250
e. $
18. The demand function for a commodity is given by
.
Find the consumer surplus when the sales level is .
19. A hot, wet summer is causing a mosquito population explosion in a lake resort area. The number of
mosquitoes is increasing at an estimated rate of per week (where t is measured in
weeks). By how much does the mosquito population increase between the 4th and th weeks of
summer?
20. The standard deviation for a random variable with probability density function f and mean µ is
defined
.
Find the standard deviation for an exponential density function with mean .
Stewart – Calculus ET 8e Chapter 8 Form E
Answer Key
Stewart – Calculus ET 8e Chapter 8 Form F
____ 1. Find the length of the curve. Select the correct answer.
a.
b. 36.05
c. 13.05
d. 25.05
e. None of these
2. Find the arc length of the graph from A to B.
(1, 2)
(4, 9)
A
B
y = x
3/2
+1
12345 x
2
4
6
8
10
12
14
16
18
20
22
24 y
3. Use differentials to approximate the arc length of the graph of the equation from P to Q. Round answer to
four decimal places.
P (1, 4), Q (1.2, 4.728)
____ 4. Find the length of the line segment joining the two given points by finding the equation of the line using
Equation (2). Then check your answer by using the distance formula. Select the correct answer.
(0, 0), (4, 6)
a.
213
b. 10
c.
10
d.
2