Stewart – Calculus ET 8e Chapter 8 Form A
1. Find the area of the surface obtained by rotating the curve about the -axis.
2. Find the coordinates of the centroid for the region bounded by the curves , x = 0,
and y = .
3. Find the centroid of the region shown, not by integration, but by locating the centroids of the
rectangles and triangles and using additivity of moments.
4. For a given commodity and pure competition, the number of units produced and the price per unit
are determined as the coordinates of the point of intersection of the supply and demand curves.
Given the demand curve
and the supply curve
,
find the consumer surplus.
5. A movie theater has been charging $ .00 per person and selling about tickets on a typical
weeknight. After surveying their customers, the theater estimates that for every $ that they lower
the price, the number of moviegoers will increase by per night. Find the demand function and
calculate the consumer surplus when the tickets are priced at $ .
Stewart – Calculus ET 8e Chapter 8 Form A
6. Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that
are normally distributed with standard deviation 12 g. If the target weight is 500 g, what is the
probability that the machine produces a box with less than g of cereal? Round your answer to
four decimal places.
7. The manager of a fast-food restaurant determines that the average time that her customers wait for
service is 2 minutes.
The manager wants to advertise that anybody who isn’t served within a certain number of minutes
gets a free hamburger. But she doesn’t want to give away free hamburgers to more than % of her
customers. What value of x must she use in the advertisement “if you aren’t served within x minutes,
you get a free hamburger”?
8. A type of lightbulb is labeled as having an average lifetime of hours. It’s reasonable to model
the probability of failure of these bulbs by an exponential density function with mean µ = .
What is the median lifetime of these lightbulbs? Give your answer rounded to two decimal places.
9. Let the function whose graph is shown be a probability density function. Calculate the mean.
10. Let
a) For what value of c is f a probability density function?
b) For that value of c, find P (–1 < X < 1).
Stewart – Calculus ET 8e Chapter 8 Form A
11. Write an integral giving the arc length of the graph of the equation from P to Q.
y = ; P (1,
1
3
), Q (3,
1
11
)
12. For the following exercise, (a) plot the graph of the function f, (b) write an integral giving the arc length of
the graph of the function over the indicated interval, and (c) find the arc length of the curve accurate to two
decimal places.
= x – 2 ; [0, 3]
13. Find the area of the surface obtained by revolving the given curve about the y-axis.
x = on [0, 2]
14. An aquarium is 3 ft long, 2 ft wide, and 2 ft deep. If the aquarium is filled with water, find the force exerted
by the water (a) on the bottom of the aquarium, (b) on the longer side of the aquarium, and (c) on the shorter
side of the aquarium. (The weight density of water is 62.4 lb/ft3.)
15. A cylindrical drum of diameter 6 ft and length 9 ft is lying on its side, submerged in water 10 ft deep. Find
the force exerted by the water on one end of the drum to the nearest pound. (The weight density of water is
62.4 lb/ft3.)
16. You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force
exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)
5 ft
7ft
9ft
17. Find the centroid of the region bounded by the graphs of the given equations.
18. Find the center of mass of the system comprising masses mk located at the points Pk in a coordinate plane.
Assume that mass is measured in grams and distance is measured in centimeters.
m1 = 2, m2 = 3, m3 = 5
P1 (–5, –1), P2 (–4, –2), P3 (4, –2)
Stewart – Calculus ET 8e Chapter 8 Form A
19. Find the centroid of the region shown in the figure.
–1
1
1–1 x
y
20. Find the center of mass of the lamina of the region shown if the density of the circular lamina is six times that
of the rectangular lamina.
2–2 x
1
–1
y
Stewart – Calculus ET 8e Chapter 8 Form A
Answer Key
Stewart – Calculus ET 8e Chapter 8 Form A
Stewart – Calculus ET 8e Chapter 8 Form B
1. Set up, but do not evaluate, an integral for the length of the curve.
2. A hawk flying at an altitude of m accidentally drops its prey. The parabolic trajectory of the falling prey
is described by the equation
until it hits the ground, where y is its height above the ground and x is the horizontal distance traveled in
meters.
True or False?
The distance traveled by the prey from the time it is dropped until the time it hits the ground is
approximately m.
3. Use the arc length formula to find the length of the curve.
4. A steady wind blows a kite due west. The kite’s height above ground from horizontal position to
ft is given by
.
Find the distance traveled by the kite.
Give your answer rounded to two decimal places.
5. Find the area of the surface obtained by rotating the curve about the -axis.
6. Find the coordinates of the centroid for the region bounded by the curves , x = 0,
and y = .
7. A swimming pool is 10 ft wide and 36 ft long and its bottom is an inclined plane, the shallow end having a
depth of ft and the deep end, 12 ft. If the pool is full of water, find the hydrostatic force on the shallow end.
(Use the fact that water weighs 62.5 lb/ .)
Stewart – Calculus ET 8e Chapter 8 Form B
8. Find the exact coordinates of the centroid.
9. Find the centroid of the region bounded by the curves.
10. Find the center of mass of a lamina in the shape of a quarter-circle with radius with density = .
11. A demand curve is given by
.
Find the consumer surplus when the selling price is $ .
Stewart – Calculus ET 8e Chapter 8 Form B
12. For a given commodity and pure competition, the number of units produced and the price per unit
are determined as the coordinates of the point of intersection of the supply and demand curves.
Given the demand curve
and the supply curve
,
find the consumer surplus.
13. A movie theater has been charging $ .00 per person and selling about tickets on a typical
weeknight. After surveying their customers, the theater estimates that for every $ that they lower
the price, the number of moviegoers will increase by per night. Find the demand function and
calculate the consumer surplus when the tickets are priced at $ .
14. The manager of a fast-food restaurant determines that the average time that her customers wait for
service is 2 minutes.
The manager wants to advertise that anybody who isn’t served within a certain number of minutes
gets a free hamburger. But she doesn’t want to give away free hamburgers to more than % of her
customers. What value of x must she use in the advertisement “if you aren’t served within x minutes,
you get a free hamburger”?
15. Let
a) For what value of c is f a probability density function?
b) For that value of c, find P (–1 < X < 1).
16. Find the area of the surface obtained by revolving the graph of y = on [0, 1] about the x-axis.
17. A cylindrical drum of diameter 2 ft and length 5 ft is lying on its side, submerged in water 17 ft deep. Find
the force exerted by the water on one end of the drum to the nearest pound. (The weight density of water is
62.4 lb/ft3.)
Stewart – Calculus ET 8e Chapter 8 Form B
18. You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force
exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)
5 ft
8ft
10 ft
19. 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)
20. Find the centroid of the region bounded by the graphs of the given equations.
Stewart – Calculus ET 8e Chapter 8 Form B
Answer Key
Stewart – Calculus ET 8e Chapter 8 Form C
Select the correct answer for each question.
____ 1. Find the length of the curve.
a.
b.
c.
d.
e.
____ 2. Use Simpson’s Rule with n = 6 to estimate the length of the curve , .
Round your answer to six decimal places.
a. 6.987208
b. 6.947368
c. 6.972089
d. 6.947582
e. None of these
Stewart – Calculus ET 8e Chapter 8 Form C
____ 3. Find the arc length of the graph from A to B.
(1, 3)
(4, 10)
A
B
y = x
3/2
+2
12345 x
2
4
6
8
10
12
14
16
18
20
22
24 y
a.
13 13
)
b.
13 13
)
c.
13 13
)
d.
13 13
)
Stewart – Calculus ET 8e Chapter 8 Form C
____ 4. Set up, but do not evaluate, an integral for the length of the curve.
a.
b.
c.
d.
e.
____ 5. Find the area of the surface obtained by rotating the circle about the line .
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 8 Form C
____ 6. 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 [4, 5]
a.
b.
c. 4
d.
4
____ 7. A gate in an irrigation canal is constructed in the form of a trapezoid ft wide at the bottom, ft wide at the
top, and 2 ft high. It is placed vertically in the canal, with the water extending to its top. Find the hydrostatic
force on one side of the gate.
a. 166 lb
b. 21 lb
c. 67lb
d. 666 lb
e. None of these
Stewart – Calculus ET 8e Chapter 8 Form C
____ 8. You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force
exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)
5 ft
6ft
a. 780 lb
b. 312 lb
c. 156 lb
d. 1,560 lb
____ 9. You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force
exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)
4ft
a. 666 lb
b. 1,331 lb
c. 2,662 lb
d. 333 lb
Stewart – Calculus ET 8e Chapter 8 Form C
____ 10. Find the centroid of the region bounded by the graphs of the given equations.
a.
b.
c.
d.
____ 11. 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.
____ 12. Find the centroid of the region bounded by the given curves.
a.
b.
c.
28
75
,
184
105
d.
e. None of these
Stewart – Calculus ET 8e Chapter 8 Form C
____ 13. Find the centroid of the region bounded by the given curves.
a.
b.
c.
d.
e.
____ 14. A rectangular tank has width 3 ft, height 2 ft, and length 7 ft. It is filled with equal volumes of water and oil.
The oil has a weight density of 50 lb/ft3 and floats on the water. Find the force exerted by the mixture on one
end of the tank. (The weight density of water is 62.4 lb/ft3.)
a. 319 lb
b. 356 lb
c. 337 lb
d. 169 lb
____ 15. 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 C
____ 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.
a. 0.029
b. 0.039
c. 0.026
d. 0.044
e.
f. 0.020
____ 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.
a. $
b. $51,367,000
c. $17,765,250
d. $26,866,667
e. $37,974,583
Stewart – Calculus ET 8e Chapter 8 Form C
____ 18. The demand function for a commodity is given by
.
Find the consumer surplus when the sales level is .
a. $9,167
b. $
c. $11,167
d. $8,167
e. $10,167
____ 19. 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.
____ 20. Suppose the average waiting time for a customer’s call to be answered by a company representative
(modeled by exponentially decreasing probability density functions) is minutes. Find the median
waiting time.
a. minutes
b. minutes
c. minutes
d. minutes
e. minutes