Question 1

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

Accepted Answer

$$\pi [ \sqrt { 2 } + \ln ( 1 + \sqrt { 2 } ) ]$$

Question 2

Use the Theorem of Pappus to find the volume of the solid obtained by revolving the region bounded by the graphs of $y = 4 - x ^ { 2 } , y = 4$, and $x = 2$ about the $y$-axis. Select the correct answer.

Accepted Answer

A)  $8 \pi$ 
B)  $\frac { 32 } { 3 } \pi$ 
C)  $4 \pi$ 
D)  $\frac { 16 } { 3 } \pi$ 
A)  $8 \pi$ 
B)  $\frac { 32 } { 3 } \pi$ 
C)  $4 \pi$ 
D)  $\frac { 16 } { 3 } \pi$ A

Question 3

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

Accepted Answer

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

Question 4

Find the exact coordinates of the centroid.
$$y = e ^ { 3 x } , y = 0 , x = 0 , x = \frac { 1 } { 3 }$$

Accepted Answer

The answer of Find the exact coordinates of the centroid.
\[y...

Question 5

A hawk flying at an altitude of $121 \mathrm {~m}$ 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 $156.532 \mathrm {~m}$.

Accepted Answer

A) True 
 B)False

Question 6

Find the length of the curve for the interval $1 \leq x \leq 4$.
$$y = \int _ { 1 } ^ { x } \sqrt { t ^ { 8 } - 1 } d t$$

Accepted Answer

The answer of Find the length of the curve for...

Question 7

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 $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$.)

Accepted Answer

The answer of You are given the shape of the...

Question 8

A type of lightbulb is labeled as having an average lifetime of 4000 hours. It's reasonable to model the probability of failure of these bulbs by an exponential density function with mean $\mu = 4000$. What is the median lifetime of these lightbulbs? Give your answer rounded to two decimal places.

Accepted Answer

The answer of A type of lightbulb is labeled as...

Question 9

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 245 .

Accepted Answer

A)  $\int _ { 40 } ^ { 195 } f ( x ) d x$ 
B)  $\int _ { 245 } ^ { 195 } f ( x ) d x$ 
C)  $\int _ { 40 } ^ { 245 } f ( x ) d x$ 
D)  $\int _ { 195 } ^ { 245 } f ( x ) d x$ 
E)  $\int _ { 0 } ^ { 245 } f ( x ) d x$ 
A)  $\int _ { 40 } ^ { 195 } f ( x ) d x$ 
B)  $\int _ { 245 } ^ { 195 } f ( x ) d x$ 
C)  $\int _ { 40 } ^ { 245 } f ( x ) d x$ 
D)  $\int _ { 195 } ^ { 245 } f ( x ) d x$ 
E)  $\int _ { 0 } ^ { 245 } f ( x ) d x$

Question 10

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 1,700 units.
Select the correct answer.

Accepted Answer

A)  $\$ 37,974,583$ 
B)  $\$ 51,367,000$ 
C)  $\$ 26,866,667$ 
D)  $\$ 17,765,250$ 
E)  $\$ 51,366,000$ 
A)  $\$ 37,974,583$ 
B)  $\$ 51,367,000$ 
C)  $\$ 26,866,667$ 
D)  $\$ 17,765,250$ 
E)  $\$ 51,366,000$

Question 11

$$\text { Let the function whose graph is shown be a probability density function. Calculate the mean. }$$

Accepted Answer

The answer of \[\text { Let the function whose graph...

Question 12

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.
$( 0,0 ) , ( 4,6 )$

Accepted Answer

A)  $2 \sqrt { 13 }$ 
B)  10 
C)  $\sqrt { 10 }$ 
D)  2 
A)  $2 \sqrt { 13 }$ 
B)  10 
C)  $\sqrt { 10 }$ 
D)  2

Question 13

Find the area of the surface obtained by rotating the circle $x ^ { 2 } + y ^ { 2 } = 8 ^ { 2 }$ about the line $y = 8$. Select the correct answer.

Accepted Answer

A)  $128 \pi ^ { 2 }$ 
B)  $256 \pi ^ { 2 }$ 
C)  $192 \pi ^ { 2 }$ 
D)  $64 \pi ^ { 2 }$ 
E)  $448 \pi ^ { 2 }$ 
A)  $128 \pi ^ { 2 }$ 
B)  $256 \pi ^ { 2 }$ 
C)  $192 \pi ^ { 2 }$ 
D)  $64 \pi ^ { 2 }$ 
E)  $448 \pi ^ { 2 }$

Question 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 $5 \%$ 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"?

Accepted Answer

The answer of The manager of a fast-food restaurant determines...

Question 15

Find the area of the surface obtained by rotating the curve about the $y$-axis.
$$x = y ^ { 3 } , 0 \leq y \leq 1$$

Accepted Answer

The answer of Find the area of the surface obtained...

Question 16

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

Accepted Answer

The answer of Find the area of the surface obtained...

Question 17

A trough has vertical ends that are equilateral triangles with sides of length $4 \mathrm { ft }$. If the trough is filled with water to a depth of $1 \mathrm { ft }$, find the force exerted by the water on one end of the trough. Round to one decimal place. (The weight density of water is $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$.) Select the correct answer.

Accepted Answer

A)  $12.0 \mathrm { lb }$ 
B)  $249.6 \mathrm { lb }$ 
C)  $6.0 \mathrm { lb }$ 
D)  $499.2 \mathrm { lb }$ 
A)  $12.0 \mathrm { lb }$ 
B)  $249.6 \mathrm { lb }$ 
C)  $6.0 \mathrm { lb }$ 
D)  $499.2 \mathrm { lb }$

Question 18

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

Accepted Answer

A)  $2.0625$ 
B)  $36.05$ 
C)  $13.05$ 
D)  $25.055$ 
E)  None of these 
A)  $2.0625$ 
B)  $36.05$ 
C)  $13.05$ 
D)  $25.055$ 
E)  None of these

Question 19

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.

Accepted Answer

The answer of Find the centroid of the region shown,...

Question 20

Find the length of the curve. Select the correct answer.
$$y = 2 \ln \left( \sin \frac { x } { 2 } \right) , \frac { \pi } { 3 } \leq x \leq \pi$$

Accepted Answer

A)  $\ln ( 2 \sqrt { 3 } )$ 
B)  $\ln ( \sqrt { 3 } )$ 
C)  $\ln ( 3 )$ 
D)  $\ln ( 2 + \sqrt { 3 } )$ 
E)  None of these 
A)  $\ln ( 2 \sqrt { 3 } )$ 
B)  $\ln ( \sqrt { 3 } )$ 
C)  $\ln ( 3 )$ 
D)  $\ln ( 2 + \sqrt { 3 } )$ 
E)  None of these

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

Use the Theorem of Pappus to find the volume of the solid obtained by revolving the region bounded by the graphs of $y = 4 - x ^ { 2 } , y = 4$ , and $x = 2$ about the $y$ -axis. Select the correct answer.

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

Find the exact coordinates of the centroid. $y = e ^ { 3 x } , y = 0 , x = 0 , x = \frac { 1 } { 3 }$

Find the length of the curve for the interval $1 \leq x \leq 4$ . $y = \int _ { 1 } ^ { x } \sqrt { t ^ { 8 } - 1 } d t$

A type of lightbulb is labeled as having an average lifetime of 4000 hours. It's reasonable to model the probability of failure of these bulbs by an exponential density function with mean $\mu = 4000$ . What is the median lifetime of these lightbulbs? Give your answer rounded to two decimal places.

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 245 .

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 1,700 units. Select the correct answer.

$\text { Let the function whose graph is shown be a probability density function. Calculate the mean. }$ $\text { Let the function whose graph is shown be a probability density function. Calculate the mean. }$

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. $( 0,0 ) , ( 4,6 )$

Find the area of the surface obtained by rotating the circle $x ^ { 2 } + y ^ { 2 } = 8 ^ { 2 }$ about the line $y = 8$ . Select the correct answer.

Find the area of the surface obtained by rotating the curve about the $y$ -axis. $x = y ^ { 3 } , 0 \leq y \leq 1$

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

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

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.

Find the length of the curve. Select the correct answer. $y = 2 \ln \left( \sin \frac { x } { 2 } \right) , \frac { \pi } { 3 } \leq x \leq \pi$

Functions and Models

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Applications of Differentiation

Integrals

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Techniques of Integration

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Vectors and the Geometry of Space

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Exam 8: Further Applications of Integration

Use the Theorem of Pappus to find the volume of the solid obtained by revolving the region bounded by the graphs of y=4−x2,y=4y = 4 - x ^ { 2 } , y = 4y=4−x2,y=4 , and x=2x = 2x=2 about the yyy -axis. Select the correct answer.

Find the area of the surface obtained by rotating the circle x2+y2=52x ^ { 2 } + y ^ { 2 } = 5 ^ { 2 }x2+y2=52 about the line y=5y = 5y=5 .

Find the exact coordinates of the centroid. y=e3x,y=0,x=0,x=13y = e ^ { 3 x } , y = 0 , x = 0 , x = \frac { 1 } { 3 }y=e3x,y=0,x=0,x=31​

Find the length of the curve for the interval 1≤x≤41 \leq x \leq 41≤x≤4 . y=∫1xt8−1dty = \int _ { 1 } ^ { x } \sqrt { t ^ { 8 } - 1 } d ty=∫1x​t8−1​dt

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 62.4lb/ft362.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }62.4lb/ft3 .)

If f(x)f ( x )f(x) is the probability density function for the blood cholesterol level of men over the age of 40 , where xxx is measured in milligrams per deciliter, express as an integral the probability that the cholesterol level of such a man lies between 195 and 245 .

Let the function whose graph is shown be a probability density function. Calculate the mean. \text { Let the function whose graph is shown be a probability density function. Calculate the mean. } Let the function whose graph is shown be a probability density function. Calculate the mean.

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)( 0,0 ) , ( 4,6 )(0,0),(4,6)

Find the area of the surface obtained by rotating the circle x2+y2=82x ^ { 2 } + y ^ { 2 } = 8 ^ { 2 }x2+y2=82 about the line y=8y = 8y=8 . Select the correct answer.

Find the area of the surface obtained by rotating the curve about the yyy -axis. x=y3,0≤y≤1x = y ^ { 3 } , 0 \leq y \leq 1x=y3,0≤y≤1

Find the area of the surface obtained by rotating the curve about the xxx -axis. x=13(y2+2)32,1≤y≤4x = \frac { 1 } { 3 } \left( y ^ { 2 } + 2 \right) ^ { \frac { 3 } { 2 } } , 1 \leq y \leq 4x=31​(y2+2)23​,1≤y≤4

Find the length of the curve. Select the correct answer. x=y48+14y2,1≤y≤2x = \frac { y ^ { 4 } } { 8 } + \frac { 1 } { 4 y ^ { 2 } } , 1 \leq y \leq 2x=8y4​+4y21​,1≤y≤2

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.

Find the length of the curve. Select the correct answer. y=2ln⁡(sin⁡x2),π3≤x≤πy = 2 \ln \left( \sin \frac { x } { 2 } \right) , \frac { \pi } { 3 } \leq x \leq \piy=2ln(sin2x​),3π​≤x≤π

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Use the Theorem of Pappus to find the volume of the solid obtained by revolving the region bounded by the graphs of $y = 4 - x ^ { 2 } , y = 4$ , and $x = 2$ about the $y$ -axis. Select the correct answer.

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

Find the exact coordinates of the centroid. $y = e ^ { 3 x } , y = 0 , x = 0 , x = \frac { 1 } { 3 }$

Find the length of the curve for the interval $1 \leq x \leq 4$ . $y = \int _ { 1 } ^ { x } \sqrt { t ^ { 8 } - 1 } d t$

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 245 .

$\text { Let the function whose graph is shown be a probability density function. Calculate the mean. }$ $\text { Let the function whose graph is shown be a probability density function. Calculate the mean. }$

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. $( 0,0 ) , ( 4,6 )$

Find the area of the surface obtained by rotating the circle $x ^ { 2 } + y ^ { 2 } = 8 ^ { 2 }$ about the line $y = 8$ . Select the correct answer.

Find the area of the surface obtained by rotating the curve about the $y$ -axis. $x = y ^ { 3 } , 0 \leq y \leq 1$

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

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

Find the length of the curve. Select the correct answer. $y = 2 \ln \left( \sin \frac { x } { 2 } \right) , \frac { \pi } { 3 } \leq x \leq \pi$