Deck 8: Further Applications of Integration

Question 1

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 $ .

Answer 1

Question 2

The manager of a fast-food restaurant determines that the average time that her customers wait for service is 2 minutes. Find the probability that a customer is served within the first 2 minutes.

A) $1.6321$
B) $0.6321$
C) $4.6321$
D) $3.6321$
E) $2.6321$

Answer 2

$0.6321$

Question 3

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"?

Answer 3

Question 4

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.

Question 5

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 $2,100+7 e^{0.7 t}$ per week (where t is measured in weeks). By how much does the mosquito population increase between the 4th and 6 th weeks of summer?

A) $4702$
B) 7702
C) 5702
D) 6702
E) 8702

Question 6

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 $250$ .

A) $\int_{40}^{250} f(x) d x$
B) $\int_{0}^{250} f(x) d x$
C) $\int_{195}^{250} f(x) d x$
D) $\int_{40}^{195} f(x) d x$
E) $\int_{250}^{195} f(x) d x$

Question 7

Let a) For what value of c is f a probability density function?
b) For that value of c, find P (-1 < X < 1).

Question 8

Let the function whose graph is shown be a probability density function. Calculate the mean.

Question 9

A demand curve is given by .
Find the consumer surplus when the selling price is $ .

Question 10

A gate in an irrigation canal is constructed in the form of a trapezoid $6$ ft wide at the bottom, $12$ 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) $1015.8$ lb
B) $998.4$ lb
C) $1009.8$ lb
D) $973$ lb
E) None of these

Question 11

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.)

A) 2662.4 lb
B) 1331.2 lb
C) 665.6 lb
D) 332.8 lb

Question 12

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.)

A) 104 lb
B) 1040 lb
C) 208 lb
D) 520 lb

Question 13

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 $20$ minutes. Find the median waiting time.

A) $16.86$ minutes
B) $17.86$ minutes
C) $13.86$ minutes
D) $14.86$ minutes
E) $15.86$ minutes

Question 14

The marginal cost function $C^{\prime}(x)$ is defined to be the derivative of the cost function. If the marginal cost of manufacturing x units of a product is $C^{\prime}(x)=0.009 x^{2}-1.8 x+9$ (measured in dollars per unit) and the fixed start-up cost is $C(0)=2,200,000$ , use the Total Change Theorem to find the cost of producing the first 5,000 units.

A) $35,454,500
B) $35,484,500
C) $35,444,500
D) $35,434,500
E) $ $35,474,500$

Question 15

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.

Question 16

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.

Question 17

The demand function for a commodity is given by $p=2000-0.1 x-0.01 x^{2}$ . Find the consumer surplus when the sales level is $125$ .

A) $10458
B) $4458
C) $ $6458$
D) $9458
E) $8458

Question 18

The standard deviation for a random variable with probability density function f and mean µ is defined $\sigma=\left[\int_{-\infty}^{\infty}(x-\mu)^{2} f(x) d x\right]^{1 / 2}$ . Find the standard deviation for an exponential density function with mean $10$ .

A) $10$
B) $2.5$
C) 2
D) $3.3$
E) 5

Question 19

For any normal distribution, find to two decimal places.

Question 20

The marginal revenue from producing x units of a certain product is $100+x-0.001 x^{2}+0.00003 x^{3}$ (in dollars per unit). Find the increase in revenue if the production level is raised from 1,100 units to $1700$ units.

A) $17765250
B) $ $51366000$
C) $26866667
D) $37974583
E) $51367000

Question 21

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.)

Question 22

A trough is filled with a liquid of density 855 kg/ $\mathrm{m}^{3}$ . The ends of the trough are equilateral triangles with sides 9 m long and vertex at the bottom. Find the hydrostatic force on one end of the trough.

A) $6.79 \times 10^{5} \mathrm{~N}$
B) $7.79 \times 10^{5} \mathrm{~N}$
C) $9.79 \times 10^{5} \mathrm{~N}$
D) $8.79 \times 10^{5} \mathrm{~N}$
E) $5.79 \times 10^{5} \mathrm{~N}$

Question 23

Find the exact coordinates of the centroid.

Question 24

Find the coordinates of the centroid for the region bounded by the curves , x = 0,
and y = .

Question 25

Find the centroid of the region bounded by the given curves. $y=4 \sin 5 x, y=0, x=0, x=\frac{\pi}{5}$

A) $(0.3146,6.2832)$
B) $(1.2565,6.2832)$
C) $(-1.2565,6.2832)$
D) $(1.2565,0.3146)$
E) $(-0.3146,0.7854)$

Question 26

An aquarium is 4 ft long, 3 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.)

Question 27

Use the Theorem of Pappus to find the volume of the solid obtained by revolving the region bounded by the graphs of $y=36-x^{2}, y=36$ and $x=6$ about the y-axis.

A) $648$ $\pi$
B) $144$ $\pi$
C) $864$ $\pi$
D) $108$ $\pi$

Question 28

Find the centroid of the region bounded by the graphs of the given equations. $y=15-x^{2}, \quad y=3-x$

A) $\left(\frac{1}{2}, \frac{37}{5}\right)$
B) $\left(\frac{1}{2}, \frac{5}{2}\right)$
C) $\left(\frac{5}{2}, \frac{1}{2}\right)$
D) $\left(\frac{37}{5}, \frac{1}{2}\right)$

Question 29

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/ .)

Question 30

Find the centroid of the region bounded by the given curves. $y=9 x^{3}, 9 x+y=18, x=0$

A) $\left(\frac{7461}{75}, \frac{261}{105}\right)$
B) $\left(\frac{252}{75}, \frac{7452}{105}\right)$
C) $\left(\frac{92}{105}, \frac{9}{75}\right)$
D) $\left(\frac{261}{75}, \frac{7461}{105}\right)$
E) None of these

Question 31

Find the center of mass of a lamina in the shape of a quarter-circle with radius with density = .

Question 32

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.) (ft)

A) 62.4 lb
B) 873.6 lb
C) 436.8 lb
D) 124.8 lb

Question 33

A rectangular tank has width 4 ft, height 4 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) 1897.6 lb
B) 1699.2 lb
C) 899.2 lb
D) 1798.4 lb

Question 34

Find the centroid of the region bounded by the graphs of the given equations. $y=|x| \sqrt{4-x^{2}}, \quad y=0, \quad x=-2, \quad x=2$

A) $\left(0, \frac{4}{5}\right)$
B) $\left(\frac{5}{4}, 0\right)$
C) $\left(\frac{4}{5}, 0\right)$
D) $\left(0, \frac{5}{4}\right)$

Question 35

Find the centroid of the region bounded by the given curves.

Question 36

The masses $m_{i}$ are located at the point $P_{i}$ . Find the moments $M_{x}$ and $M_{y}$ and the center of mass of the system. $m_{1}=3, m_{2}=7, m_{3}=113$ ; $p_{1}(1,5), P_{2}(3,-2), p_{3}(-2,-1)$

A) $M_{x}=22, M_{y}=50,\left(\frac{50}{23}, \frac{22}{23}\right)$
B) $M_{x}=50, M_{y}=22,\left(\frac{50}{23}, \frac{22}{23}\right)$
C) $M_{x}=-50, M_{y}=22,\left(\frac{50}{23},-\frac{22}{23}\right)$
D) $M_{x}=22, M_{y}=50,\left(\frac{22}{23}, \frac{50}{23}\right)$
E) $M_{x}=50, M_{y}=-22,\left(-\frac{50}{23}, \frac{22}{23}\right)$

Question 37

Calculate the center of mass of the lamina with density = .

Question 38

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.

Question 39

Find the centroid of the region bounded by the curves.

Question 40

A trough has vertical ends that are equilateral triangles with sides of length 2 ft. If the trough is filled with water to a depth of 1 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.)

A) 31.2 lb
B) 12.0 lb
C) 62.4 lb
D) 6.0 lb

Question 41

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.)

Question 42

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.) (ft)

Question 43

Find the centroid of the region shown in the figure.

Question 44

Find the volume obtained when the circle of radius 2 with center ( 2 , 0) is rotated about the y-axis.

A) $\frac{32}{3}$ $\pi$
B) $32$ $\pi$
C) $2 \pi$
D) $\frac{3}{32} \pi$
E) $\pi$

Question 45

Find the area of the surface obtained by rotating the circle $x^{2}+y^{2}=5^{2}$ about the line $y=5$ .

A) $25 \pi^{2}$
B) $75 \pi^{2}$
C) $175 \pi^{2}$
D) $50 \pi^{2}$
E) $100 \pi^{2}$

Question 46

Find the area of the surface obtained by revolving the given curve about the x-axis. $y=\frac{e^{x}+e^{-x}}{2}$ on $[0, \ln 5]$

A)
$\frac{156}{25} \pi$
B) $\frac{\pi}{5}\left(\frac{156}{25}+\ln 5\right)$
C) $\pi\left(\frac{156}{25}+\ln 5\right)$
D) $\pi \ln 5$

Question 47

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.) (ft)

Question 48

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve $y=\ln x^{7}$ about the x-axis on the interval $1 \leq x \leq 7$ .

A) $\int_{7}^{1} 2 \pi \sqrt{1+\left(\frac{7}{x}\right)^{2}} d x$
B) $\int_{7}^{1} 2 \pi x \ln (x) \sqrt{1+\left(\frac{7}{x}\right)^{2}} d x$
C) $\int_{1}^{7} 2 \pi(7 x) \sqrt{1+\left(\frac{7}{x}\right)^{2}} d x$
D) $\int_{7}^{1} 2 \pi(7 x) \sqrt{1+\left(\frac{7}{x}\right)^{2}} d x$
E) $\int_{1}^{7} 2 \pi(7 \ln (x)) \sqrt{1+\left(\frac{7}{x}\right)^{2}} d x$

Question 49

Find the centroid of the region bounded by the graphs of and .

Question 50

A cylindrical drum of diameter 2 ft and length 6 ft is lying on its side, submerged in water 16 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.)

Question 51

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.)

Question 52

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve about the given axis. $y=e^{x}, 1 \leq y \leq 9 ;$ y-axis

A) $\int_{1}^{9} 2 \pi x \sqrt{1+e^{2 x}} d x$
B) $\int_{1}^{9} 2 \pi x e^{9 x} \sqrt{1+e^{9 x}} d x$
C) $\int_{0}^{\ln 9} 2 \pi x \sqrt{1+e^{2 x}} d x$
D) $\int_{1}^{9} 2 \pi \sqrt{1+e^{2 x}} d x$
E) $\int_{0}^{\ln 9} 2 \pi x \sqrt{1+e^{9 x}} d x$

Question 53

Find the area of the surface obtained by rotating the curve about the y-axis. $y=\frac{1}{4} x^{2}-\frac{1}{2} \ln x, 1 \leq x \leq 8$

A) $\frac{3 \pi}{2}$
B) $\frac{10}{3}$
C) $\frac{8 \pi}{3}$
D) $\frac{11 \pi}{2}$
E) None of these

Question 54

Find the area of the surface obtained by rotating the curve about the x-axis. $x=\frac{1}{3}\left(y^{2}+2\right)^{\frac{3}{2}}, 1 \leq y \leq 4$

A) $\frac{285}{2}$ $\pi$
B) $283$ $\pi$
C) $281$ $\pi$
D) $\frac{287 \pi}{2}$
E) $\frac{289 \pi}{2}$

Question 55

Write an integral giving the area of the surface obtained by revolving the curve about the x-axis. (Do not evaluate the integral.) y = $\frac{4}{x}$ on [3, 6]

A) ( )
B) 8 $\int_{3}^{6} x^{3} \sqrt{x^{4}+16} d x$
C) 8 $\int_{3}^{6} \frac{\sqrt{x^{4}+16}}{x^{3}} d x$
D) ( $\int_{3}^{6} \frac{4}{x}\left(1+\left(-\frac{4}{x^{2}}\right)^{2}\right) d x$ )

Question 56

Find the area of the region under the graph of f on [a, b]. $f(x)=x^{2}-2 x+2 ; \quad[-1,2]$

A) -3
B) -6
C) 6
D) 3

Question 57

Find the centroid of the region bounded by the graphs of the given equations.

Question 58

Find the center of mass of the lamina of the region shown if the density of the circular lamina is five times that of the rectangular lamina.

Question 59

Find the centroid of the region bounded by the graphs of and .

Question 60

Find the center of mass of the system comprising masses m_k located at the points P_k in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters.
m₁ = 3, m₂ = 4, m₃ = 5
P₁ (-3, 5), P₂ (3, 4), P₃ (-4, 1)

Question 61

Find the length of the curve for the interval .

Question 62

Find the area of the surface obtained by revolving the given curve about the y-axis.
x = on [0, 2]

Question 63

Find the area of the surface obtained by rotating the curve about the -axis.

Question 64

Set up, but do not evaluate, an integral that represents the length of the curve.

Question 65

Use differentials to approximate the arc length of the graph of the equation from P to Q. Round answer to four decimal places. $y=x^{2}+3$ P (3, 12), Q (3.2, 13.24)

A) 6.0828
B) 3.6497
C) 1.2166
D) 2.4331

Question 66

Find the arc length of the graph from A to B.

A) $\frac{1}{54}(\sqrt{37}-$ $80 \sqrt{10}$ )
B) $\frac{1}{108}(\sqrt{37}-$ $80 \sqrt{10}$ )
C) $\frac{1}{54}(296 \sqrt{37}-$ $80 \sqrt{10}$ )
D) $\frac{1}{108}(296 \sqrt{37}-$ $80 \sqrt{10}$ )

Question 67

Set up, but do not evaluate, an integral for the length of the curve.

Question 68

Find the length of the curve. $y=\frac{1}{6}\left(x^{2}+4\right)^{3 / 2}, 0 \leq x \leq 3$

A) $6.5000$
B) $7.5000$
C) $8.5000$
D) $5.5000$
E) $4.5000$

Question 69

Find the length of the curve. $x=\frac{y^{4}}{8}+\frac{1}{4 y^{2}}, 1 \leq y \leq 3$

A) 13.05
B) 25.05 5
C) $10.2225$
D) 36.05
E) None of these

Question 70

Find the length of the curve for the interval $1 \leq x \leq 4$ . $y=\int_{1}^{x} \sqrt{t^{4}-1} d t$

A) $L=15.7500$
B) $L=10.5000$
C) $L=12.6000$
D) $L=21.0000$
E) $L=0.0476$

Question 71

Find the length of the curve. $y=2 \ln \left(\sin \frac{x}{2}\right), \frac{\pi}{5} \leq x \leq \pi$

A) $\ln (2+\sqrt{5})$
B) $\ln (\sqrt{5})$
C) $\ln (5)$
D) $\ln (2 \sqrt{5})$
E) None of these

Question 72

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. (0, 0), (1, 8)

A) $\sqrt{62}$
B) 9
C) $\sqrt{65}$
D) 3

Question 73

Find the area of the surface obtained by revolving the graph of y = on [0, 1] about the x-axis.

Question 74

Find the arc length of the graph of the given equation on the specified interval. y = $\frac{2}{3}$ ( $x^{2}$ + 1) $3 / 2$
, [2, 5]

A) $81$
B) $\frac{243}{2}$
C) $\frac{81}{2}$
D) $\frac{243}{4}$

Question 75

Find the area of the surface obtained by revolving the given curve about the x-axis. on [0, 1]

Question 76

If the infinite curve $y=e^{x}, x \geq 0$ , is rotated about the x-axis , find the area of the resulting surface.

A) $\frac{\pi}{6}[\sqrt{2}+\ln (1+\sqrt{2})]$
B) $\frac{\pi}{6}[\sqrt{3}+\ln (1+\sqrt{3})]$
C) $\frac{\pi}{3}[\sqrt{2}+\ln (1+\sqrt{2})]$
D) $\pi[\sqrt{2}+\ln (1+\sqrt{2})]$
E) $\frac{\pi}{4}[\sqrt{2}+\ln (1+\sqrt{2})]$

Question 77

Use Simpson's Rule with n = 6 to estimate the length of the curve $y=3 \sin x$ , $0 \leq x \leq \pi$ . Round your answer to six decimal places.

A) 6.987208
B) 6.947368
C) 6.972089
D) 6.947582
E) None of these

Question 78

Set up, but do not evaluate, an integral for the length of the curve. $y=2 e^{x} \sin x, \quad \quad 0 \leq x \leq \frac{9 \pi}{2}$

A) $L=\int_{0}^{9 x / 2} \sqrt{1-4 e^{2 x}(1+\sin 2 x)} d x$
B) $L=\int_{0}^{9 x / 2} \sqrt{1+4 e^{2 x}(1-\sin x)} d x$
C) $L=\int_{0}^{9 x / 2} \sqrt{1-4 e^{2 x}(1-\sin 2 x)} d x$
D) $L=\int_{0}^{9 x / 2} \sqrt{1+4 e^{2 x}(1-\sin 2 x)} d x$
E) $L=\int_{0}^{9 x / 2} \sqrt{1+4 e^{2 x}(1+\sin 2 x)} d x$

Question 79

Find the arc length function for the curve $y=10 x^{3 / 2}$ with starting point $P_{0}(1,15)$ .

A) $L=\frac{2}{675}\left((1+225 x)^{3 / 2}+226\right)$
B) $L=\frac{2}{225}\left((1+220 x)^{3 / 2}-226 \sqrt{226}\right)$
C) $L=\frac{2}{675}\left((1+225 x)^{5 / 2}+226 \sqrt{226}\right)$
D) $L=\frac{2}{675}\left((1+225 x)^{3 / 2}-226 \sqrt{226}\right)$
E) $L=\frac{2}{675}\left((1+225 x)^{1 / 2}-226 \sqrt{226}\right)$

Question 80

A hawk flying at an altitude of $121$ m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation $y=121-\frac{x^{2}}{64}$ 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 $156.532$ m.