Question 1

Find the center of mass of a thin plate of constant density covering the given region.
-The region bounded by the parabola $y = 16 - x ^ { 2 }$ and the $x$-axis

Accepted Answer

A)  $\bar { x } = 0 , \bar { y } = \frac { 5 } { 32 }$ 
B)  $\bar { x } = \frac { 32 } { 5 } , \bar { y } = 0$ 
C)  $\bar { x } = 0 , \bar { y } = \frac { 32 } { 5 }$ 
D)  $\bar { x } = 0 , \bar { y } = \frac { 8192 } { 15 }$ 
A)  $\bar { x } = 0 , \bar { y } = \frac { 5 } { 32 }$ 
B)  $\bar { x } = \frac { 32 } { 5 } , \bar { y } = 0$ 
C)  $\bar { x } = 0 , \bar { y } = \frac { 32 } { 5 }$ 
D)  $\bar { x } = 0 , \bar { y } = \frac { 8192 } { 15 }$

Question 2

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 $\mathrm { y } = 4 \sqrt { \mathrm { x } } , \mathrm { y } = 4$, and $\mathrm { x } = 0$ about the line $\mathrm { x } = 1$

Accepted Answer

A)  $\frac { 68 } { 15 } \pi$ 
B)  $\frac { 32 } { 15 } \pi$ 
C)  $\frac { 14 } { 15 } \pi$ 
D)  $\frac { 28 } { 15 } \pi$ 
A)  $\frac { 68 } { 15 } \pi$ 
B)  $\frac { 32 } { 15 } \pi$ 
C)  $\frac { 14 } { 15 } \pi$ 
D)  $\frac { 28 } { 15 } \pi$

Question 3

Find the center of mass of a thin plate covering the given region with the given density function.
-The region bounded below by the parabola $y = x ^ { 2 }$ and above by the line $y = x + 2$, with density $\delta ( x ) = 7 x ^ { 2 }$

Accepted Answer

A)  $\bar { x } = \frac { 18 } { 5 } , \bar { y } = \frac { 531 } { 70 }$ 
B)  $\bar { x } = \frac { 8 } { 7 } , \bar { y } = \frac { 118 } { 49 }$ 
C)  $\bar { x } = \frac { 531 } { 70 } , \bar { y } = \frac { 18 } { 5 }$ 
D)  $\bar { x } = \frac { 9 } { 8 } , \bar { y } = \frac { 131 } { 64 }$ 
A)  $\bar { x } = \frac { 18 } { 5 } , \bar { y } = \frac { 531 } { 70 }$ 
B)  $\bar { x } = \frac { 8 } { 7 } , \bar { y } = \frac { 118 } { 49 }$ 
C)  $\bar { x } = \frac { 531 } { 70 } , \bar { y } = \frac { 18 } { 5 }$ 
D)  $\bar { x } = \frac { 9 } { 8 } , \bar { y } = \frac { 131 } { 64 }$

Question 4

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 $y = \frac { x ^ { 3 } } { 3 }$ from $x = 0$ to $x = \sqrt [ 4 ] { 3 }$

Accepted Answer

A)  $\frac { 1 } { 12 }$ 
B)  $\frac { 7 } { 18 }$ 
C)  $\frac { 7 } { 12 }$ 
D)  $\frac { 3 } { 4 }$ 
A)  $\frac { 1 } { 12 }$ 
B)  $\frac { 7 } { 18 }$ 
C)  $\frac { 7 } { 12 }$ 
D)  $\frac { 3 } { 4 }$

Question 5

Find the volume of the solid generated by revolving the shaded region about the given axis.
-About the $x$-axis

$y = 16 - x ^ { 2 }$

Accepted Answer

A)  $\frac { 2048 } { 15 } \pi$ 
B)  $\frac { 64 } { 3 } \pi$ 
C)  $\frac { 7168 } { 15 } \pi$ 
D)  $\frac { 8192 } { 15 } \pi$ 
A)  $\frac { 2048 } { 15 } \pi$ 
B)  $\frac { 64 } { 3 } \pi$ 
C)  $\frac { 7168 } { 15 } \pi$ 
D)  $\frac { 8192 } { 15 } \pi$

Question 6

Find the volume of the described solid.
-The solid lies between planes perpendicular to the $x$-axis at $x = 0$ and $x = 9$. The cross sections perpendicular to the $x$-axis between these planes are squares whose bases run from the parabola $y = - 2 \sqrt { x }$ to the parabola $y = 2 \sqrt { x }$.

Accepted Answer

A)  162 
B)  648 
C)  324 
D)  640 
A)  162 
B)  648 
C)  324 
D)  640

Question 7

Solve the problem.
-The hemispherical bowl of radius 5 contains water to a depth 3. Find the volume of water in the bowl.

Accepted Answer

A)  $36 \pi$ 
B)  $\frac { 358 } { 3 } \pi$ 
C)  $\frac { 233 } { 3 } \pi$ 
D)  $18 \pi$ 
A)  $36 \pi$ 
B)  $\frac { 358 } { 3 } \pi$ 
C)  $\frac { 233 } { 3 } \pi$ 
D)  $18 \pi$

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 $y = 2$

Accepted Answer

A)  $\frac { 20 } { 7 } \pi$ 
B)  $\frac { 55 } { 7 } \pi$ 
C)  $\frac { 110 } { 7 } \pi$ 
D)  $6 \pi$ 
A)  $\frac { 20 } { 7 } \pi$ 
B)  $\frac { 55 } { 7 } \pi$ 
C)  $\frac { 110 } { 7 } \pi$ 
D)  $6 \pi$

Question 9

Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis.
-$$y = x ^ { 3 } , 0 \leq x \leq 2 ; x \text {-axis }$$

Accepted Answer

A)  $2 \pi \int _ { 0 } ^ { 2 } x ^ { 3 } \sqrt { 1 + 9 x ^ { 4 } } d x$ 
B)  $\pi \int _ { 0 } ^ { 2 } x ^ { 2 } \sqrt { 1 + 9 x ^ { 4 } } d x$ 
C)  $\pi \int _ { 0 } ^ { 2 } x ^ { 3 } \sqrt { 1 + 9 x ^ { 4 } } d x$ 
D)  $2 \pi \int _ { 0 } ^ { 2 } x ^ { 2 } \sqrt { 1 + 9 x ^ { 4 } } d x$ 
A)  $2 \pi \int _ { 0 } ^ { 2 } x ^ { 3 } \sqrt { 1 + 9 x ^ { 4 } } d x$ 
B)  $\pi \int _ { 0 } ^ { 2 } x ^ { 2 } \sqrt { 1 + 9 x ^ { 4 } } d x$ 
C)  $\pi \int _ { 0 } ^ { 2 } x ^ { 3 } \sqrt { 1 + 9 x ^ { 4 } } d x$ 
D)  $2 \pi \int _ { 0 } ^ { 2 } x ^ { 2 } \sqrt { 1 + 9 x ^ { 4 } } d x$

Question 10

Solve.
-Find the surface area of the cone frustum generated by revolving the line segment $y = ( x / 2 ) + ( 1 / 2 ) , 1 \leq x \leq 3$, about the $y$-axis.

Accepted Answer

A)  $12 \pi \sqrt { 5 }$ 
B)  $15 \pi \sqrt { 5 }$ 
C)  $10 \pi \sqrt { 5 }$ 
D)  $11 \pi \sqrt { 5 }$ 
A)  $12 \pi \sqrt { 5 }$ 
B)  $15 \pi \sqrt { 5 }$ 
C)  $10 \pi \sqrt { 5 }$ 
D)  $11 \pi \sqrt { 5 }$

Question 11

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.
-$$y = \frac { 6 } { x } , y = 0 , x = 7 , x = 9$$

Accepted Answer

A)  $36 \pi$ 
B)  $24 \pi$ 
C)  $18 \pi$ 
D)  $12 \pi$ 
A)  $36 \pi$ 
B)  $24 \pi$ 
C)  $18 \pi$ 
D)  $12 \pi$

Question 12

Set up an integral for the length of the curve.
-$$y = \int _ { 0 } ^ { x } \cot t d t , \frac { \pi } { 6 } \leq x \leq \frac { \pi } { 3 }$$

Accepted Answer

A)  $\int _ { \pi / 6 } ^ { \pi / 3 } \csc x d x$ 
B)  $\int _ { \pi / 6 } ^ { \pi / 3 } \sqrt { 1 + \cot x } d x$ 
C)  $\int _ { \pi / 6 } ^ { \pi / 3 } \sqrt { 1 + \csc ^ { 4 } x } d x$ 
D)  $\int _ { \pi / 6 } ^ { \pi / 3 } \sqrt { \csc x } d x$ 
A)  $\int _ { \pi / 6 } ^ { \pi / 3 } \csc x d x$ 
B)  $\int _ { \pi / 6 } ^ { \pi / 3 } \sqrt { 1 + \cot x } d x$ 
C)  $\int _ { \pi / 6 } ^ { \pi / 3 } \sqrt { 1 + \csc ^ { 4 } x } d x$ 
D)  $\int _ { \pi / 6 } ^ { \pi / 3 } \sqrt { \csc x } d x$

Question 13

Find the area of the surface generated by revolving the curve about the indicated axis.
-$x = 3 \sqrt { 4 - y } , 0 \leq y \leq 15 / 4$; y-axis

Accepted Answer

A)  $\frac { 125 } { 2 } \pi$ 
B)  $\left( \frac { 125 } { 2 } - 5 \sqrt { 10 } \right) \pi$ 
C)  $5 \pi \sqrt { 10 }$ 
D)  $\left( \frac { 125 } { 2 } + 5 \sqrt { 10 } \right) \pi$ 
A)  $\frac { 125 } { 2 } \pi$ 
B)  $\left( \frac { 125 } { 2 } - 5 \sqrt { 10 } \right) \pi$ 
C)  $5 \pi \sqrt { 10 }$ 
D)  $\left( \frac { 125 } { 2 } + 5 \sqrt { 10 } \right) \pi$

Question 14

Provide an appropriate response.
-The region shown here is to be revolved about the x-axis to generate a solid. Which of the methods (disk, washer, shell) could you use to find the volume of the solid? How many integrals would be required in each case?
 
$x = 2 y - y ^ { 2 }$

Accepted Answer

The volume can be found using

Question 15

Solve the problem.
-An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the curve $y = 1 - \frac { x ^ { 2 } } { 16 } , - 4$ $\leq x \leq 4$, about the $x$-axis (dimensions are in feet). How many cubic feet of fuel will the tank hold to the nearest cubic foot?

Accepted Answer

A)  17 
B)  4 
C)  13 
D)  7 
A)  17 
B)  4 
C)  13 
D)  7

Question 16

Find the length of the curve.
-$$y = \int _ { 0 } ^ { x } \sqrt { 25 \sin ^ { 2 } t - 1 } d t , 0 \leq x \leq \frac { \pi } { 2 }$$

Accepted Answer

A)  5 
B)  $\frac { 5 } { 2 }$ 
C)  25 
D)  $\frac { 5 } { 3 }$ 
A)  5 
B)  $\frac { 5 } { 2 }$ 
C)  25 
D)  $\frac { 5 } { 3 }$

Question 17

Find the center of mass of a thin plate of constant density covering the given region.
-The region bounded by $y = 8 - x$ and the axes

Accepted Answer

A)  $\bar { x } = \frac { 8 } { 3 } , \bar { y } = \frac { 8 } { 3 }$ 
B)  $\bar { x } = 8 , \bar { y } = 8$ 
C)  $\bar { x } = 32 , \bar { y } = 32$ 
D)  $\bar { x } = \frac { 256 } { 3 } , \bar { y } = \frac { 256 } { 3 }$ 
A)  $\bar { x } = \frac { 8 } { 3 } , \bar { y } = \frac { 8 } { 3 }$ 
B)  $\bar { x } = 8 , \bar { y } = 8$ 
C)  $\bar { x } = 32 , \bar { y } = 32$ 
D)  $\bar { x } = \frac { 256 } { 3 } , \bar { y } = \frac { 256 } { 3 }$

Question 18

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
-$$y = \frac { 2 } { x } , y = - x + 3$$

Accepted Answer

A)  $\frac { 1 } { 4 } \pi$ 
B)  $\frac { 1 } { 3 } \pi$ 
C)  $9 \pi$ 
D)  $2 \pi$ 
A)  $\frac { 1 } { 4 } \pi$ 
B)  $\frac { 1 } { 3 } \pi$ 
C)  $9 \pi$ 
D)  $2 \pi$

Question 19

Set up an integral for the length of the curve.
-$x = y ^ { 1 / 7 } , 0 \leq y \leq 2$

Accepted Answer

A)  $\int _ { 0 } ^ { 2 } \sqrt { \frac { 1 } { 49 y ^ { 12 / 7 } } } d y$ 
B)  $\int _ { 0 } ^ { 2 } \sqrt { \frac { 2 } { 49 y ^ { 12 / 7 } } } d y$ 
C)  $\int _ { 0 } ^ { 2 } \sqrt { \frac { 7 y ^ { 6 / 7 } + 1 } { 7 y ^ { 6 / 7 } } } d y$ 
D)  $\int _ { 0 } ^ { 2 } \sqrt { \frac { 49 \mathrm { y } ^ { 12 / 7 } + 1 } { 49 \mathrm { y } 12 / 7 } } d y$ 
A)  $\int _ { 0 } ^ { 2 } \sqrt { \frac { 1 } { 49 y ^ { 12 / 7 } } } d y$ 
B)  $\int _ { 0 } ^ { 2 } \sqrt { \frac { 2 } { 49 y ^ { 12 / 7 } } } d y$ 
C)  $\int _ { 0 } ^ { 2 } \sqrt { \frac { 7 y ^ { 6 / 7 } + 1 } { 7 y ^ { 6 / 7 } } } d y$ 
D)  $\int _ { 0 } ^ { 2 } \sqrt { \frac { 49 \mathrm { y } ^ { 12 / 7 } + 1 } { 49 \mathrm { y } 12 / 7 } } d y$

Question 20

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.
-$$x = 18 - y ^ { 2 } , x = y ^ { 2 } , y = 0$$

Accepted Answer

A)  $81 \pi$ 
B)  $\frac { 81 } { 4 } \pi$ 
C)  $\frac { 81 } { 2 } \pi$ 
D)  $162 \pi$ 
A)  $81 \pi$ 
B)  $\frac { 81 } { 4 } \pi$ 
C)  $\frac { 81 } { 2 } \pi$ 
D)  $162 \pi$

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by the parabola $y = 16 - x ^ { 2 }$ and the $x$ -axis

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 $\mathrm { y } = 4 \sqrt { \mathrm { x } } , \mathrm { y } = 4$ , and $\mathrm { x } = 0$ about the line $\mathrm { x } = 1$

Find the center of mass of a thin plate covering the given region with the given density function. -The region bounded below by the parabola $y = x ^ { 2 }$ and above by the line $y = x + 2$ , with density $\delta ( x ) = 7 x ^ { 2 }$

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 $y = \frac { x ^ { 3 } } { 3 }$ from $x = 0$ to $x = \sqrt [ 4 ] { 3 }$

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the $x$ -axis $Find the volume of the solid generated by revolving the shaded region about the given axis. -About the x -axis y = 16 - x ^ { 2 }$ $y = 16 - x ^ { 2 }$

Solve the problem. -The hemispherical bowl of radius 5 contains water to a depth 3. Find the volume of water in the bowl.

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line. -About the line $y = 2$

Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. - $y = x ^ { 3 } , 0 \leq x \leq 2 ; x \text {-axis }$

Solve. -Find the surface area of the cone frustum generated by revolving the line segment $y = ( x / 2 ) + ( 1 / 2 ) , 1 \leq x \leq 3$ , about the $y$ -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. - $y = \frac { 6 } { x } , y = 0 , x = 7 , x = 9$

Set up an integral for the length of the curve. - $y = \int _ { 0 } ^ { x } \cot t d t , \frac { \pi } { 6 } \leq x \leq \frac { \pi } { 3 }$

Find the area of the surface generated by revolving the curve about the indicated axis. - $x = 3 \sqrt { 4 - y } , 0 \leq y \leq 15 / 4$ ; y-axis

Solve the problem. -An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the curve $y = 1 - \frac { x ^ { 2 } } { 16 } , - 4$ $\leq x \leq 4$ , about the $x$ -axis (dimensions are in feet). How many cubic feet of fuel will the tank hold to the nearest cubic foot?

Find the length of the curve. - $y = \int _ { 0 } ^ { x } \sqrt { 25 \sin ^ { 2 } t - 1 } d t , 0 \leq x \leq \frac { \pi } { 2 }$

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by $y = 8 - x$ and the axes

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = \frac { 2 } { x } , y = - x + 3$

Set up an integral for the length of the curve. - $x = y ^ { 1 / 7 } , 0 \leq y \leq 2$

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. - $x = 18 - y ^ { 2 } , x = y ^ { 2 } , y = 0$

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Exam 7: Applications of Definite Integrals

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by the parabola y=16−x2y = 16 - x ^ { 2 }y=16−x2 and the xxx -axis

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 y=4x,y=4\mathrm { y } = 4 \sqrt { \mathrm { x } } , \mathrm { y } = 4y=4x​,y=4 , and x=0\mathrm { x } = 0x=0 about the line x=1\mathrm { x } = 1x=1

Find the center of mass of a thin plate covering the given region with the given density function. -The region bounded below by the parabola y=x2y = x ^ { 2 }y=x2 and above by the line y=x+2y = x + 2y=x+2 , with density δ(x)=7x2\delta ( x ) = 7 x ^ { 2 }δ(x)=7x2

Find the moment or center of mass of the wire, as indicated. -Find the moment about the xxx -axis of a wire of constant density that lies along the curve y=x33y = \frac { x ^ { 3 } } { 3 }y=3x3​ from x=0x = 0x=0 to x=34x = \sqrt [ 4 ] { 3 }x=43​

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the xxx -axis y=16−x2y = 16 - x ^ { 2 }y=16−x2