Deck 8: Further Applications of Integration

Question 1

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.

Answer 1

Question 2

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

Answer 2

Question 3

For the following exercise, (a) plot the graph of the function (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.

Answer 3

a. b. c.3.71

Question 4

A type of lightbulb is labeled as having an average lifetime of It's reasonable to model the probability of failure of these bulbs by an exponential density function with mean What is the median lifetime of these lightbulbs? Give yourAnswer rounded to two decimal places.

Question 5

Find the area of the surface obtained by revolving the given curve about the

Question 6

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 7

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 )

Question 8

Let a) For what value of is probability density function? b) For that value of , find

Question 9

Find the center of mass of the system comprising masses located at the points in a coordinate plane.Assume that mass is measured in grams and distance is measured in centimeters.

Question 10

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

Question 11

Write an integral giving the arc length of the graph of the equation from to

Question 12

Find the centroid of the region shown in the figure.

Question 13

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 )

Question 14

Find the coordinates of the centroid for the region bounded by the curves and

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 of cereal? Round yourAnswer to four decimal places.

Question 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 .)

Question 17

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 18

A movie theater has been charging 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

Question 19

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 f her customers.What value of x must she use in the advertisement "if you aren't served within minutes, you get a free hamburger"?

Question 20

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

Question 21

Find the coordinates of the centroid for the region bounded by the curves , and

Question 22

A hawk flying at an altitude of accidentally drops its prey.The parabolic trajectory of the falling prey is described by the equation until it hits the ground, where is its height above the ground and 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

Question 23

Find the centroid of the region bounded by the curves.

Question 24

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

Question 25

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

Question 26

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

Question 27

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

Question 28

a) For what value of is a probability density function? b) For that value of find

Question 29

Find the area of the surface obtained by rotating the curve about the

Question 30

Find the area of the surface obtained by revolving the graph of about the

Question 31

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 32

Use the arc length formula to find the length of the curve.

Question 33

A vertical plate is submerged in water (the surface of the water coincides with the Find the force exerted by the water on the plate.(The weight density of water is

Question 34

A movie theater has been charging per person and selling about 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 ind the demand function and calculate the consumer surplus when the tickets are priced at

Question 35

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 must she use in the advertisement "if you aren't served within minutes, you get a free hamburger"?

Question 36

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 )

Question 37

Find the exact coordinates of the centroid.

Question 38

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

Question 39

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 )

Question 40

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 yourAnswer rounded to two decimal places.

Question 41

Select the correct Answer: for each question.
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

A)666 lb
B)1,331 lb
C)2,662 lb
D)333 lb

Question 42

Select the correct Answer: for each question.
The standard deviation for a random variable with probability density function and mean is defined Find the standard deviation for an exponential density function with mean

A)
B)
C)
D)
E)

Question 43

Select the correct Answer: for each question.
Set up, but do not evaluate, an integral for the length of the curve.

A)
B)
C)
D)
E)

Question 44

Select the correct Answer: for each question.
Suppose the average waiting time for a customer's call to beAnswered 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

Question 45

Select the correct Answer: for each question.
Find the length of the curve.

A)
B)
C)
D)
E)

Question 46

Select the correct Answer: for each question.
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

Question 47

Select the correct Answer: for each question.
Find the arc length of the graph from A to B.

A)
B)
C)
D)

Question 48

Select the correct Answer: for each question.
Find the centroid of the region bounded by the given curves.

A)
B)
C)
D)
E)

Question 49

Select the correct Answer: for each question.
Write an integral giving the area of the surface obtained by revolving the curve about the x-axis.(Do not evaluate the integral.)

A)
B)
C)
D)

Question 50

Select the correct Answer: for each question.
A trough is filled with a liquid of density .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)

Question 51

Select the correct Answer: for each question.
Find the centroid of the region bounded by the graphs of the given equations.

A)
B)
C)
D)

Question 52

Select the correct Answer: for each question.
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

A)780 lb
B)312 lb
C)156 lb
D)1,560 lb

Question 53

Select the correct Answer: for each question.
The marginal revenue from producing units of a certain product is 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

Question 54

Select the correct Answer: for each question.
Find the area of the surface obtained by rotating the circle about the line

A)
B)
C)
D)
E)

Question 55

Select the correct Answer: for each question.
Dye dilution is a method of measuring cardiac output.If mg of dye is used and c(t) is the concentration of the dye at time then the cardiac output over the time interval is given by Find the cardiac output over the time interval [0, 15] if the dye dilution method is used with of dye and the dye concentration, in mg/L, is modeled by where is measured in seconds.

A)0.029
B)0.039
C)0.026
D)0.044
E)
F)0.020

Question 56

Select the correct Answer: for each question.
Find the centroid of the region bounded by the graphs of the given equations.

A)
B)
C)
D)

Question 57

Select the correct Answer: for each question.
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

Question 58

Select the correct Answer: for each question.
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 and floats on the water.Find the force exerted by the mixture on one end of the tank.(The weight density of water is )

A)319 lb
B)356 lb
C)337 lb
D)169 lb

Question 59

Select the correct Answer: for each question.
Find the centroid of the region bounded by the given curves.

A)
B)
C)
D)
E)None of these

Question 60

Select the correct Answer: for each question.
Use Simpson's Rule with o estimate the length of the curve Round yourAnswer to six decimal places.

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

Question 61

Select the correct Answer for each question.
Find the area of the surface obtained by rotating the circle about the line

A)
B)
C)
D)
E)

Question 62

Select the correct Answer for each question.
The masses are located at the points Find the moments and and the center of mass of the system.

A)
B)
C)
D)
E)

Question 63

Select the correct Answer for each question.
If 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)

Question 64

Select the correct Answer for each question.
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)

Question 65

Select the correct Answer for each question.
Write an integral giving the area of the surface obtained by revolving the curve about the (Do not evaluate the integral.)

A)
B)
C)
D)

Question 66

Select the correct Answer for each question.
Find the centroid of the region bounded by the given curves.

A)
B)
C)
D)
E)

Question 67

Select the correct Answer for each question.
Find the length of the curve.

A)
B)
C)
D)
E)

Question 68

Select the correct Answer for each question.
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 )

A)624 lb
B)62 lb
C)125 lb
D)312 lb

Question 69

Select the correct Answer for each question.
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)

Question 70

Select the correct Answer for each question.
Find the length of the curve.

A)
B)
C)
D)
E)None of these

Question 71

Select the correct Answer for each question.
The marginal revenue from producing x units of a certain product is 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

Question 72

Select the correct Answer for each question.
Find the arc length of the graph of the given equation on the specified interval.y =

A)
B)
C)
D)

Question 73

Select the correct Answer for each question.
Find the centroid of the region bounded by the graphs of the given equations.

A)
B)
C)
D)

Question 74

Select the correct Answer for each question.
Set up, but do not evaluate, an integral for the length of the curve.

A)
B)
C)
D)
E)

Question 75

Select the correct Answer for each question.
Find the area of the surface obtained by revolving the given curve about the x-axis.

A)
B)
C)
D)

Question 76

Select the correct Answer for each question.
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

Question 77

Select the correct Answer for each question.
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)

Question 78

Select the correct Answer for each question.
Dye is a method of measuring cardiac output.If mg of dye is used and is the concentration of the dye at time t, then the cardiac output over the time interval 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 concentra odeled by where is measured in seconds.

A)0.031
B)0.043
C)0.029
D)0.049
E)
F)0.022

Question 79

Select the correct Answer for each question.
A trough is filled with a liquid of density .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)

Question 80

Select the correct Answer for each question.
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)
B)
C)
D)