Question 1

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

Accepted Answer

$$30 - 10 \sqrt { 3 } \approx 12.67$$

Question 2

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve $y = \ln x ^ { 5 }$ about the $x$-axis on the interval $1 \leq x \leq 5$.

Accepted Answer

A) 
$$\int _ { 5 } ^ { 1 } 2 \pi x \ln ( x ) \sqrt { 1 + \left( \frac { 5 } { x } \right) ^ { 2 } } d x$$ 
B) 
$$\int _ { 5 } ^ { 1 } 2 \pi \sqrt { 1 + \left( \frac { 5 } { x } \right) ^ { 2 } } d x$$ 
C) 
$$\int _ { 1 } ^ { 5 } 2 \pi ( 5 \ln ( x ) ) \sqrt { 1 + \left( \frac { 5 } { x } \right) ^ { 2 } } d x$$ 
D) 
$$\int _ { 5 } ^ { 1 } 2 \pi ( 5 x ) \sqrt { 1 + \left( \frac { 5 } { x } \right) ^ { 2 } } d x$$ 
E) 
$$\int _ { 1 } ^ { 5 } 2 \pi ( 5 x ) \sqrt { 1 + \left( \frac { 5 } { x } \right) ^ { 2 } } d x$$ 
A) 
$$\int _ { 5 } ^ { 1 } 2 \pi x \ln ( x ) \sqrt { 1 + \left( \frac { 5 } { x } \right) ^ { 2 } } d x$$ 
B) 
$$\int _ { 5 } ^ { 1 } 2 \pi \sqrt { 1 + \left( \frac { 5 } { x } \right) ^ { 2 } } d x$$ 
C) 
$$\int _ { 1 } ^ { 5 } 2 \pi ( 5 \ln ( x ) ) \sqrt { 1 + \left( \frac { 5 } { x } \right) ^ { 2 } } d x$$ 
D) 
$$\int _ { 5 } ^ { 1 } 2 \pi ( 5 x ) \sqrt { 1 + \left( \frac { 5 } { x } \right) ^ { 2 } } d x$$ 
E) 
$$\int _ { 1 } ^ { 5 } 2 \pi ( 5 x ) \sqrt { 1 + \left( \frac { 5 } { x } \right) ^ { 2 } } d x$$ C

Question 3

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.
$$m _ { 1 } = 2 , \quad m _ { 2 } = 3 , \quad m _ { 3 } = 5$$
$$P _ { 1 } ( - 5 , - 1 ) , \quad P _ { 2 } ( - 4 , - 2 ) , \quad P _ { 3 } ( 4 , - 2 )$$

Accepted Answer

$$\left( - \frac { 1 } { 5 } , - \frac { 9 } { 5 } \right)$$

Question 4

Find the length of the curve.
$$y = 2 \ln \left( \sin \frac { x } { 2 } \right) , \frac { \pi } { 4 } \leq x \leq \pi$$

Accepted Answer

A)  $\ln ( 4 )$ 
B)  $\ln ( 2 + \sqrt { 4 } )$ 
C)  $\ln ( 2 \sqrt { 4 } )$ 
D)  $\ln ( \sqrt { 4 } )$ 
E)  None of these 
A)  $\ln ( 4 )$ 
B)  $\ln ( 2 + \sqrt { 4 } )$ 
C)  $\ln ( 2 \sqrt { 4 } )$ 
D)  $\ln ( \sqrt { 4 } )$ 
E)  None of these

Question 5

Set up, but do not evaluate, an integral for the length of the curve.
$$y = 6 e ^ { x } \sin x , \quad 0 \leq x \leq \frac { 9 \pi } { 2 }$$

Accepted Answer

A) 
$L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 - 36 e ^ { 2 x } ( 1 + \sin 2 x ) } d x$ 
B) 
$L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 + 36 e ^ { 2 x } ( 1 + \sin 2 x ) } d x$ 
C) 
$L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 - 36 e ^ { 2 x } ( 1 - \sin 2 x ) } d x$ 
D) 
$L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 + 36 e ^ { 2 x } ( 1 - \sin 2 x ) } d x$ 
E) 
$L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 + 36 e ^ { 2 x } ( 1 - \sin x ) } d x$ 
A) 
$L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 - 36 e ^ { 2 x } ( 1 + \sin 2 x ) } d x$ 
B) 
$L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 + 36 e ^ { 2 x } ( 1 + \sin 2 x ) } d x$ 
C) 
$L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 - 36 e ^ { 2 x } ( 1 - \sin 2 x ) } d x$ 
D) 
$L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 + 36 e ^ { 2 x } ( 1 - \sin 2 x ) } d x$ 
E) 
$L = \int _ { 0 } ^ { 9 x / 2 } \sqrt { 1 + 36 e ^ { 2 x } ( 1 - \sin x ) } d x$

Question 6

A cylindrical drum of diameter $6 \mathrm { ft }$ and length $9 \mathrm { ft }$ is lying on its side, submerged in water $10 \mathrm { ft }$ deep. Find the force exerted by the water on one end of the drum to the nearest pound. (The weight density of water is $\left. 62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 } . \right)$

Accepted Answer

The answer of A cylindrical drum of diameter \(6 \mathrm...

Question 7

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 8

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)  $50 \pi ^ { 2 }$

Question 9

Boxes are labeled as containing $500 \mathrm {~g}$ of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation $12 \mathrm {~g}$. If the target weight is $500 \mathrm {~g}$, what is the probability that the machine produces a box with less than $490 \mathrm {~g}$ of cereal? Round your answer to four decimal places.

Accepted Answer

The answer of Boxes are labeled as containing \(500 \mathrm...

Question 10

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

A)  $666 \mathrm { lb }$ 
B)  $1,331 \mathrm { lb }$ 
C)  $2,662 \mathrm { lb }$ 
D)  $333 \mathrm { lb }$ 
A)  $666 \mathrm { lb }$ 
B)  $1,331 \mathrm { lb }$ 
C)  $2,662 \mathrm { lb }$ 
D)  $333 \mathrm { lb }$

Question 11

An aquarium is $3 \mathrm { ft }$ long, $2 \mathrm { ft }$ wide, and $2 \mathrm { ft }$ 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 $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$.)

Accepted Answer

The answer of An aquarium is \(3 \mathrm { ft...

Question 12

A trough is filled with a liquid of density $855 \mathrm {~kg} / \mathrm { m } ^ { 3 }$. The ends of the trough are equilateral triangles with sides $7 \mathrm {~m}$ long and vertex at the bottom. Find the hydrostatic force on one end of the trough.

Accepted Answer

A)  $2.5 \times 10 ^ { 5 } \mathrm {~N}$ 
B)  $5.5 \times 10 ^ { 5 } \mathrm {~N}$ 
C)  $4.5 \times 10 ^ { 5 } \mathrm {~N}$ 
D)  $6.5 \times 10 ^ { 5 } \mathrm {~N}$ 
E)  $3.5 \times 10 ^ { 5 } \mathrm {~N}$ 
A)  $2.5 \times 10 ^ { 5 } \mathrm {~N}$ 
B)  $5.5 \times 10 ^ { 5 } \mathrm {~N}$ 
C)  $4.5 \times 10 ^ { 5 } \mathrm {~N}$ 
D)  $6.5 \times 10 ^ { 5 } \mathrm {~N}$ 
E)  $3.5 \times 10 ^ { 5 } \mathrm {~N}$

Question 13

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 14

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 15

Find the coordinates of the centroid for the region bounded by the curves $y = 4 x ^ { 2 } , x = 0$, and $y = 144$.

Accepted Answer

The answer of Find the coordinates of the centroid for...

Question 16

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

Accepted Answer

A)  $624 \mathrm { lb }$ 
B)  $62 \mathrm { lb }$ 
C)  $125 \mathrm { lb }$ 
D)  $312 \mathrm { lb }$ 
A)  $624 \mathrm { lb }$ 
B)  $62 \mathrm { lb }$ 
C)  $125 \mathrm { lb }$ 
D)  $312 \mathrm { lb }$

Question 17

Write an integral giving the arc length of the graph of the equation from $\mathbf { P }$ to $\mathbf { Q }$.
$$y = \frac { 1 } { x ^ { 2 } + 2 } ; \mathrm { P } \left( 1 , \frac { 1 } { 3 } \right) , \mathrm { Q } \left( 3 , \frac { 1 } { 11 } \right)$$

Accepted Answer

The answer of Write an integral giving the arc length...

Question 18

Set up, but do not evaluate, an integral for the length of the curve.
$$y = 2 e ^ { x } \sin x , \quad 0 \leq x \leq \frac { 3 \pi } { 2 }$$

Accepted Answer

A) 
$L = \int _ { 0 } ^ { 3 x / 2 } \sqrt { 1 - 4 e ^ { 2 x } ( 1 + \sin 2 x ) } d x$ 
B) 
$L = \int _ { 0 } ^ { 3 x / 2 } \sqrt { 1 + 4 e ^ { 2 x } ( 1 - \sin x ) } d x$ 
C) 
$L = \int _ { 0 } ^ { 3 x / 2 } \sqrt { 1 + 4 e ^ { 2 x } ( 1 - \sin 2 x ) } d x$ 
D) 
$L = \int _ { 0 } ^ { 3 x / 2 } \sqrt { 1 - 4 e ^ { 2 x } ( 1 - \sin 2 x ) } d x$ 
E) 
$L = \int _ { 0 } ^ { 3 x / 2 } \sqrt { 1 + 4 e ^ { 2 x } ( 1 + \sin 2 x ) } d x$ 
A) 
$L = \int _ { 0 } ^ { 3 x / 2 } \sqrt { 1 - 4 e ^ { 2 x } ( 1 + \sin 2 x ) } d x$ 
B) 
$L = \int _ { 0 } ^ { 3 x / 2 } \sqrt { 1 + 4 e ^ { 2 x } ( 1 - \sin x ) } d x$ 
C) 
$L = \int _ { 0 } ^ { 3 x / 2 } \sqrt { 1 + 4 e ^ { 2 x } ( 1 - \sin 2 x ) } d x$ 
D) 
$L = \int _ { 0 } ^ { 3 x / 2 } \sqrt { 1 - 4 e ^ { 2 x } ( 1 - \sin 2 x ) } d x$ 
E) 
$L = \int _ { 0 } ^ { 3 x / 2 } \sqrt { 1 + 4 e ^ { 2 x } ( 1 + \sin 2 x ) } d x$

Question 19

The demand function for a commodity is given by
$p = 2,000 - 0.1 x - 0.01 x ^ { 2 }$ Find the consumer surplus when the sales level is 100 .

Accepted Answer

A)  $\$ 9,167$ 
B)  $\$ 7,167$ 
C)  $\$ 11,167$ 
D)  $\$ 8,167$ 
E)  $\$ 10,167$ 
A)  $\$ 9,167$ 
B)  $\$ 7,167$ 
C)  $\$ 11,167$ 
D)  $\$ 8,167$ 
E)  $\$ 10,167$

Question 20

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 71 inches and 75 inches tall?

Accepted Answer

A)  $0.759$ 
B)  $0.5222$ 
C)  $0.2812$ 
D)  $0.6512$ 
E)  $0.7154$
F) $0.5528$ 
A)  $0.759$ 
B)  $0.5222$ 
C)  $0.2812$ 
D)  $0.6512$ 
E)  $0.7154$
F) $0.5528$

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

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve $y = \ln x ^ { 5 }$ about the $x$ -axis on the interval $1 \leq x \leq 5$ .

Find the length of the curve. $y = 2 \ln \left( \sin \frac { x } { 2 } \right) , \frac { \pi } { 4 } \leq x \leq \pi$

Set up, but do not evaluate, an integral for the length of the curve. $y = 6 e ^ { x } \sin x , \quad 0 \leq x \leq \frac { 9 \pi } { 2 }$

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

A trough is filled with a liquid of density $855 \mathrm {~kg} / \mathrm { m } ^ { 3 }$ . The ends of the trough are equilateral triangles with sides $7 \mathrm {~m}$ long and vertex at the bottom. Find the hydrostatic force on one end of the trough.

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 coordinates of the centroid for the region bounded by the curves $y = 4 x ^ { 2 } , x = 0$ , and $y = 144$ .

Write an integral giving the arc length of the graph of the equation from $\mathbf { P }$ to $\mathbf { Q }$ . $y = \frac { 1 } { x ^ { 2 } + 2 } ; \mathrm { P } \left( 1 , \frac { 1 } { 3 } \right) , \mathrm { Q } \left( 3 , \frac { 1 } { 11 } \right)$

Set up, but do not evaluate, an integral for the length of the curve. $y = 2 e ^ { x } \sin x , \quad 0 \leq x \leq \frac { 3 \pi } { 2 }$

The demand function for a commodity is given by $p = 2,000 - 0.1 x - 0.01 x ^ { 2 }$ Find the consumer surplus when the sales level is 100 .

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 71 inches and 75 inches tall?

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

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Exam 8: Further Applications of Integration

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

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve y=ln⁡x5y = \ln x ^ { 5 }y=lnx5 about the xxx -axis on the interval 1≤x≤51 \leq x \leq 51≤x≤5 .

Find the length of the curve. y=2ln⁡(sin⁡x2),π4≤x≤πy = 2 \ln \left( \sin \frac { x } { 2 } \right) , \frac { \pi } { 4 } \leq x \leq \piy=2ln(sin2x​),4π​≤x≤π

Set up, but do not evaluate, an integral for the length of the curve. y=6exsin⁡x,0≤x≤9π2y = 6 e ^ { x } \sin x , \quad 0 \leq x \leq \frac { 9 \pi } { 2 }y=6exsinx,0≤x≤29π​

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 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 .

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 .)

A trough is filled with a liquid of density 855 kg/m3855 \mathrm {~kg} / \mathrm { m } ^ { 3 }855 kg/m3 . The ends of the trough are equilateral triangles with sides 7 m7 \mathrm {~m}7 m long and vertex at the bottom. Find the hydrostatic force on one end of the trough.

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 coordinates of the centroid for the region bounded by the curves y=4x2,x=0y = 4 x ^ { 2 } , x = 0y=4x2,x=0 , and y=144y = 144y=144 .

A vertical plate is submerged in water (the surface of the water coincides with the xxx -axis). Find the force exerted by the water on the plate. (The weight density of water is 62.4lb/ft362.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }62.4lb/ft3 .)

Set up, but do not evaluate, an integral for the length of the curve. y=2exsin⁡x,0≤x≤3π2y = 2 e ^ { x } \sin x , \quad 0 \leq x \leq \frac { 3 \pi } { 2 }y=2exsinx,0≤x≤23π​

The demand function for a commodity is given by p=2,000−0.1x−0.01x2p = 2,000 - 0.1 x - 0.01 x ^ { 2 }p=2,000−0.1x−0.01x2 Find the consumer surplus when the sales level is 100 .

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 2.82.82.8 inches. What is the probability that an adult male chosen at random is between 71 inches and 75 inches tall?

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

Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve $y = \ln x ^ { 5 }$ about the $x$ -axis on the interval $1 \leq x \leq 5$ .

Find the length of the curve. $y = 2 \ln \left( \sin \frac { x } { 2 } \right) , \frac { \pi } { 4 } \leq x \leq \pi$

Set up, but do not evaluate, an integral for the length of the curve. $y = 6 e ^ { x } \sin x , \quad 0 \leq x \leq \frac { 9 \pi } { 2 }$

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

A trough is filled with a liquid of density $855 \mathrm {~kg} / \mathrm { m } ^ { 3 }$ . The ends of the trough are equilateral triangles with sides $7 \mathrm {~m}$ long and vertex at the bottom. Find the hydrostatic force on one end of the trough.

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 coordinates of the centroid for the region bounded by the curves $y = 4 x ^ { 2 } , x = 0$ , and $y = 144$ .

Set up, but do not evaluate, an integral for the length of the curve. $y = 2 e ^ { x } \sin x , \quad 0 \leq x \leq \frac { 3 \pi } { 2 }$

The demand function for a commodity is given by $p = 2,000 - 0.1 x - 0.01 x ^ { 2 }$ Find the consumer surplus when the sales level is 100 .

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 71 inches and 75 inches tall?