Question 1

Find the length of the curve.
-$$x = \frac { 2 } { 3 } ( y - 1 ) ^ { 3 / 2 } \text { from } y = 16 \text { to } y = 25$$

Accepted Answer

A)  $\frac { 109 } { 3 }$ 
B)  $\frac { 183 } { 2 }$ 
C)  $\frac { 122 } { 3 }$ 
D)  61 
E)  4 
A)  $\frac { 109 } { 3 }$ 
B)  $\frac { 183 } { 2 }$ 
C)  $\frac { 122 } { 3 }$ 
D)  61 
E)  4

Question 2

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and
lines about the x-axis.
-$$x = \frac { 1 } { 6 } e ^ { 2 } , y = 0 , x = 0 , y = 1$$

Accepted Answer

A)  $3 \pi \mathrm { e }$ 
B)  $\pi ( \mathrm { e } - 1 )$ 
C)  $\frac { \pi } { 6 } ( e - 1 )$ 
D)  $6 \pi ( e - 1 )$ 
A)  $3 \pi \mathrm { e }$ 
B)  $\pi ( \mathrm { e } - 1 )$ 
C)  $\frac { \pi } { 6 } ( e - 1 )$ 
D)  $6 \pi ( e - 1 )$

Question 3

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis.
-About the $y$-axis

$y = \frac { 4 \sin ( x ) } { x } ; 0 < x \leq \pi$

Accepted Answer

A)  $20 \pi$ 
B)  $8 \pi$ 
C)  $16 \pi$ 
D)  $12 \pi$ 
A)  $20 \pi$ 
B)  $8 \pi$ 
C)  $16 \pi$ 
D)  $12 \pi$

Question 4

Solve the problem.
-A fisherman is about to reel in a 9-lb fish located 12 ft directly below him. If the fishing line weighs 1 oz per foot, how much work will it take to reel in the fish? Round your answer to the nearest tenth, if necessary.

Accepted Answer

A)  120 ft · lb 
B)  180 ft · lb 
C)  117 ft · lb 
D)  112.5 ft · lb 
A)  120 ft · lb 
B)  180 ft · lb 
C)  117 ft · lb 
D)  112.5 ft · lb

Question 5

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line.
-About the line $y = 5$

$$x = 5 y - y ^ { 2 }$$

Accepted Answer

A)  $\frac { 625 } { 3 } \pi$ 
B)  $\frac { 625 } { 12 } \pi$ 
C)  $\frac { 625 } { 6 } \pi$ 
D)  $\frac { 125 } { 6 } \pi$ 
A)  $\frac { 625 } { 3 } \pi$ 
B)  $\frac { 625 } { 12 } \pi$ 
C)  $\frac { 625 } { 6 } \pi$ 
D)  $\frac { 125 } { 6 } \pi$

Question 6

Set up an integral for the length of the curve.
-$$y = 6 \cos x , 0 \leq x \leq \pi$$

Accepted Answer

A)  $\int _ { 0 } ^ { \pi } \sqrt { 1 + 6 \sin x } d x$ 
B)  $\int _ { 0 } ^ { \pi } \sqrt { 1 - 6 \sin x } d x$ 
C)  $\int _ { 0 } ^ { \pi } \sqrt { 1 + 36 \cos ^ { 2 } x } d x$ 
D)  $\int _ { 0 } ^ { \pi } \sqrt { 1 + 36 \sin ^ { 2 } x } d x$ 
A)  $\int _ { 0 } ^ { \pi } \sqrt { 1 + 6 \sin x } d x$ 
B)  $\int _ { 0 } ^ { \pi } \sqrt { 1 - 6 \sin x } d x$ 
C)  $\int _ { 0 } ^ { \pi } \sqrt { 1 + 36 \cos ^ { 2 } x } d x$ 
D)  $\int _ { 0 } ^ { \pi } \sqrt { 1 + 36 \sin ^ { 2 } x } d x$

Question 7

Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method.
-The region bounded by $x = 3 \sqrt { y } , x = - 3 y$, and $y = 1$ about the line $y = 1$

Accepted Answer

A)  $3 \pi$ 
B)  $\frac { 11 } { 5 } \pi$ 
C)  $\frac { 22 } { 5 } \pi$ 
D)  $\frac { 13 } { 5 } \pi$ 
A)  $3 \pi$ 
B)  $\frac { 11 } { 5 } \pi$ 
C)  $\frac { 22 } { 5 } \pi$ 
D)  $\frac { 13 } { 5 } \pi$

Question 8

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line.
-About the line $y = 3$

$x = y + 6$
$x = y ^ { 2 }$

Accepted Answer

A)  $\frac { 63 } { 2 } \pi$ 
B)  $\frac { 99 } { 2 } \pi$ 
C)  $99 \pi$ 
D)  $\frac { 225 } { 2 } \pi$ 
A)  $\frac { 63 } { 2 } \pi$ 
B)  $\frac { 99 } { 2 } \pi$ 
C)  $99 \pi$ 
D)  $\frac { 225 } { 2 } \pi$

Question 9

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and
lines about the x-axis.
-$$y = \sqrt { x } , y = 0 , y = x - 6$$

Accepted Answer

A)  $27 \pi$ 
B)  $\frac { 63 } { 4 } \pi$ 
C)  $\frac { 225 } { 2 } \pi$ 
D)  $\frac { 63 } { 2 } \pi$ 
A)  $27 \pi$ 
B)  $\frac { 63 } { 4 } \pi$ 
C)  $\frac { 225 } { 2 } \pi$ 
D)  $\frac { 63 } { 2 } \pi$

Question 10

Find the volume of the described solid.
-The base of a solid is the region between the curve $y = 5 \cos x$ and the $x$-axis from $x = 0$ to $x = \pi / 2$. The cross sections perpendicular to the $x$-axis are squares with diagonals running from the $x$-axis to the curve.

Accepted Answer

A)  $\frac { 25 } { 4 } \pi$ 
B)  $\frac { 5 } { 2 } \pi$ 
C)  $6 \pi$ 
D)  $\frac { 25 } { 8 } \pi$ 
A)  $\frac { 25 } { 4 } \pi$ 
B)  $\frac { 5 } { 2 } \pi$ 
C)  $6 \pi$ 
D)  $\frac { 25 } { 8 } \pi$

Question 11

Find the moment or center of mass of the wire, as indicated.
-Find the moment about the $x$-axis of a wire of constant density that lies along the curve $y = 2 \sqrt { x }$ from $x = 0$ to $x = 3$.

Accepted Answer

A)  $\frac { 28 } { 3 }$ 
B)  21 
C)  14 
D)  $\frac { 14 } { 3 }$ 
A)  $\frac { 28 } { 3 }$ 
B)  21 
C)  14 
D)  $\frac { 14 } { 3 }$

Question 12

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and
lines about the x-axis.
-y = 3x, y = 6x, y = 3

Accepted Answer

A)  $\frac { 9 } { 2 } \pi$ 
B)  $9 \pi$ 
C)  $3 \pi$ 
D)  $\frac { 3 } { 2 } \pi$ 
A)  $\frac { 9 } { 2 } \pi$ 
B)  $9 \pi$ 
C)  $3 \pi$ 
D)  $\frac { 3 } { 2 } \pi$

Question 13

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis.
-$y = 8 - x ^ { 2 } , y = x ^ { 2 } , x = 0$

Accepted Answer

A)  $32 \pi$ 
B)  $16 \pi$ 
C)  $8 \pi$ 
D)  $4 \pi$ 
A)  $32 \pi$ 
B)  $16 \pi$ 
C)  $8 \pi$ 
D)  $4 \pi$

Question 14

Find the volume of the solid generated by revolving the region about the given line.
-The region in the second quadrant bounded above by the curve $y = 4 - x ^ { 2 }$, below by the $x$-axis, and on the right by the $y$-axis, about the line $x = 1$

Accepted Answer

A)  $\frac { 256 } { 15 } \pi$ 
B)  $8 \pi$ 
C)  $\frac { 56 } { 3 } \pi$ 
D)  $\frac { 32 } { 3 } \pi$ 
A)  $\frac { 256 } { 15 } \pi$ 
B)  $8 \pi$ 
C)  $\frac { 56 } { 3 } \pi$ 
D)  $\frac { 32 } { 3 } \pi$

Question 15

Find the volume of the solid generated by revolving the region about the y-axis.
-The region enclosed by $x = \frac { 5 } { y } , x = 0 , y = 1 , y = 2$

Accepted Answer

A)  $\frac { 25 } { 2 } \pi$ 
B)  $\frac { 5 } { 2 } \pi$ 
C)  $\frac { 25 } { 4 } \pi$ 
D)  $\frac { 75 } { 2 } \pi$ 
A)  $\frac { 25 } { 2 } \pi$ 
B)  $\frac { 5 } { 2 } \pi$ 
C)  $\frac { 25 } { 4 } \pi$ 
D)  $\frac { 75 } { 2 } \pi$

Question 16

Solve the problem.
-A company applies a clear glaze finish on the outside of the ceramic bowls it produces. The bowl corresponds to the bottom half of a sphere which is created by rotating the circle $x ^ { 2 } + y ^ { 2 } = 36$ around the $x$-axis. The finish is to be $0.2 \mathrm {~cm}$ thick, and the company wants to create 3000 bowls. Use the fact that $1 \mathrm {~L} = 1000 \mathrm {~cm} ^ { 3 }$ to calculate how many liters of finish are required. Assume that all specifications for the bowl are in $\mathrm { cm }$.

Accepted Answer

A)  2.26 L of finish 
B)  226.19 L of finish 
C)  67.86 L of finish 
D)  135.72 L of finish 
A)  2.26 L of finish 
B)  226.19 L of finish 
C)  67.86 L of finish 
D)  135.72 L of finish

Question 17

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and
lines about the x-axis.
-$$x = 8 y ^ { 2 } , x = 8 \sqrt [ 3 ] { y }$$

Accepted Answer

A)  $\frac { 10 } { 7 } \pi$ 
B)  $\frac { 8 } { 3 } \pi$ 
C)  $\frac { 20 } { 7 } \pi$ 
D)  $12 \pi$ 
A)  $\frac { 10 } { 7 } \pi$ 
B)  $\frac { 8 } { 3 } \pi$ 
C)  $\frac { 20 } { 7 } \pi$ 
D)  $12 \pi$

Question 18

Find the area of the surface generated by revolving the curve about the indicated axis.
-$y = x ^ { 3 } / 9,0 \leq x \leq 2$; $x$-axis

Accepted Answer

A)  $\frac { 98 } { 81 } \pi$ 
B)  $\frac { 256 } { 27 } \pi$ 
C)  $98 \pi$ 
D)  $\frac { 1163 } { 2187 } \pi$ 
A)  $\frac { 98 } { 81 } \pi$ 
B)  $\frac { 256 } { 27 } \pi$ 
C)  $98 \pi$ 
D)  $\frac { 1163 } { 2187 } \pi$

Question 19

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis.
-$$y = 7 e ^ { x ^ { 2 } } , y = 0 , x = 0 , x = 1$$

Accepted Answer

A)  $14 \mathrm { e } \pi$ 
B)  $7 ( \mathrm { e } - 1 ) \pi$ 
C)  $14 ( \mathrm { e } - 1 ) \pi$ 
D)  $\frac { 7 } { 2 } ( \mathrm { e } - 1 ) \pi$ 
A)  $14 \mathrm { e } \pi$ 
B)  $7 ( \mathrm { e } - 1 ) \pi$ 
C)  $14 ( \mathrm { e } - 1 ) \pi$ 
D)  $\frac { 7 } { 2 } ( \mathrm { e } - 1 ) \pi$

Question 20

Solve.
-Find the lateral surface area of the cone generated by revolving the line segment $y = x / 4,0 \leq x \leq 5$ about the $y$-axis.

Accepted Answer

A)  $20 \pi \sqrt { 19 }$ 
B)  $20 \pi \sqrt { 17 }$ 
C)  $100 \pi \sqrt { 19 }$ 
D)  $100 \pi \sqrt { 17 }$ 
A)  $20 \pi \sqrt { 19 }$ 
B)  $20 \pi \sqrt { 17 }$ 
C)  $100 \pi \sqrt { 19 }$ 
D)  $100 \pi \sqrt { 17 }$

Find the length of the curve. - $x = \frac { 2 } { 3 } ( y - 1 ) ^ { 3 / 2 } \text { from } y = 16 \text { to } y = 25$

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. - $x = \frac { 1 } { 6 } e ^ { 2 } , y = 0 , x = 0 , y = 1$

Solve the problem. -A fisherman is about to reel in a 9-lb fish located 12 ft directly below him. If the fishing line weighs 1 oz per foot, how much work will it take to reel in the fish? Round your answer to the nearest tenth, if necessary.

Set up an integral for the length of the curve. - $y = 6 \cos x , 0 \leq x \leq \pi$

Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. -The region bounded by $x = 3 \sqrt { y } , x = - 3 y$ , and $y = 1$ about the line $y = 1$

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. - $y = \sqrt { x } , y = 0 , y = x - 6$

Find the volume of the described solid. -The base of a solid is the region between the curve $y = 5 \cos x$ and the $x$ -axis from $x = 0$ to $x = \pi / 2$ . The cross sections perpendicular to the $x$ -axis are squares with diagonals running from the $x$ -axis to the curve.

Find the moment or center of mass of the wire, as indicated. -Find the moment about the $x$ -axis of a wire of constant density that lies along the curve $y = 2 \sqrt { x }$ from $x = 0$ to $x = 3$ .

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. -y = 3x, y = 6x, y = 3

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. - $y = 8 - x ^ { 2 } , y = x ^ { 2 } , x = 0$

Find the volume of the solid generated by revolving the region about the given line. -The region in the second quadrant bounded above by the curve $y = 4 - x ^ { 2 }$ , below by the $x$ -axis, and on the right by the $y$ -axis, about the line $x = 1$

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by $x = \frac { 5 } { y } , x = 0 , y = 1 , y = 2$

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. - $x = 8 y ^ { 2 } , x = 8 \sqrt [ 3 ] { y }$

Find the area of the surface generated by revolving the curve about the indicated axis. - $y = x ^ { 3 } / 9,0 \leq x \leq 2$ ; $x$ -axis

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. - $y = 7 e ^ { x ^ { 2 } } , y = 0 , x = 0 , x = 1$

Solve. -Find the lateral surface area of the cone generated by revolving the line segment $y = x / 4,0 \leq x \leq 5$ about the $y$ -axis.

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Exam 7: Applications of Definite Integrals

Find the length of the curve. - x=23(y−1)3/2 from y=16 to y=25x = \frac { 2 } { 3 } ( y - 1 ) ^ { 3 / 2 } \text { from } y = 16 \text { to } y = 25x=32​(y−1)3/2 from y=16 to y=25

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. - x=16e2,y=0,x=0,y=1x = \frac { 1 } { 6 } e ^ { 2 } , y = 0 , x = 0 , y = 1x=61​e2,y=0,x=0,y=1

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. -About the yyy -axis y=4sin⁡(x)x;0<x≤πy = \frac { 4 \sin ( x ) } { x } ; 0 < x \leq \piy=x4sin(x)​;0<x≤π