Question 1

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
-$$y = x ^ { 2 } , y = 0 , x = 0 , x = 5$$

Accepted Answer

A)  $\frac { 125 } { 3 } \pi$ 
B)  $625 \pi$ 
C)  $\frac { 3125 } { 4 } \pi$ 
D)  $\frac { 625 } { 4 } \pi$ 
A)  $\frac { 125 } { 3 } \pi$ 
B)  $625 \pi$ 
C)  $\frac { 3125 } { 4 } \pi$ 
D)  $\frac { 625 } { 4 } \pi$

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 y-axis.
-$$y = x ^ { 2 } - 5 , y = 4 x , x = 0 \text {, for } x \geq 0$$

Accepted Answer

A)  $\frac { 875 } { 6 } \pi$ 
B)  $125 \pi$ 
C)  $\frac { 875 } { 3 } \pi$ 
D)  $\frac { 875 } { 12 } \pi$ 
A)  $\frac { 875 } { 6 } \pi$ 
B)  $125 \pi$ 
C)  $\frac { 875 } { 3 } \pi$ 
D)  $\frac { 875 } { 12 } \pi$

Question 3

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves
about the given lines.
-$y = 4 x , \quad y = 0 , \quad x = 3$; revolve about the $x$-axis

Accepted Answer

A)  $288 \pi$ 
B)  $72 \pi$ 
C)  $144 \pi$ 
D)  72 
A)  $288 \pi$ 
B)  $72 \pi$ 
C)  $144 \pi$ 
D)  72

Question 4

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
-$$y = \frac { 1 } { x } , y = 0 , x = 1 , x = 9$$

Accepted Answer

A)  $\frac { 4 } { 9 } \pi$ 
B)  $\frac { 8 } { 9 } \pi$ 
C)  $\frac { 1 } { 3 } \pi$ 
D)  $\pi \ln 9$ 
A)  $\frac { 4 } { 9 } \pi$ 
B)  $\frac { 8 } { 9 } \pi$ 
C)  $\frac { 1 } { 3 } \pi$ 
D)  $\pi \ln 9$

Question 5

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

Accepted Answer

A)  $2 \pi ^ { 2 } - 2 \pi$ 
B)  $2 \pi ^ { 2 } + 4 \pi$ 
C)  $2 \pi ^ { 2 }$ 
D)  $2 \pi ^ { 2 } - 4 \pi$ 
A)  $2 \pi ^ { 2 } - 2 \pi$ 
B)  $2 \pi ^ { 2 } + 4 \pi$ 
C)  $2 \pi ^ { 2 }$ 
D)  $2 \pi ^ { 2 } - 4 \pi$

Question 6

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
-$$y = 3 \cos \pi x , y = 0 , x = - 0.5 , x = 0.5$$

Accepted Answer

A)  $\frac { 9 } { 2 } \pi$ 
B)  $18 \pi$ 
C)  $9 \pi$ 
D)  $3 \pi$ 
A)  $\frac { 9 } { 2 } \pi$ 
B)  $18 \pi$ 
C)  $9 \pi$ 
D)  $3 \pi$

Question 7

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.
-$$x = 3 y ^ { 2 } , x = - 3 y , y = 3$$

Accepted Answer

A)  $351 \pi$ 
B)  $\frac { 351 } { 2 } \pi$ 
C)  $\frac { 351 } { 8 } \pi$ 
D)  $\frac { 351 } { 4 } \pi$ 
A)  $351 \pi$ 
B)  $\frac { 351 } { 2 } \pi$ 
C)  $\frac { 351 } { 8 } \pi$ 
D)  $\frac { 351 } { 4 } \pi$

Question 8

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.
-$$y = 3 x , y = 6 x , x = 3$$

Accepted Answer

A)  $54 \pi$ 
B)  $27 \pi$ 
C)  $\frac { 243 } { 2 } \pi$ 
D)  $18 \pi$ 
A)  $54 \pi$ 
B)  $27 \pi$ 
C)  $\frac { 243 } { 2 } \pi$ 
D)  $18 \pi$

Question 9

Find the volume of the solid generated by revolving the region about the given line.
-The region in the first quadrant bounded above by the line $5 x + y = 10$, below by the $x$-axis, and on the left by the $y$-axis, about the line $x = - 2$

Accepted Answer

A)  $\frac { 320 } { 3 } \pi$ 
B)  $\frac { 200 } { 3 } \pi$ 
C)  $\frac { 160 } { 3 } \pi$ 
D)  $\frac { 25 } { 3 } \pi$ 
A)  $\frac { 320 } { 3 } \pi$ 
B)  $\frac { 200 } { 3 } \pi$ 
C)  $\frac { 160 } { 3 } \pi$ 
D)  $\frac { 25 } { 3 } \pi$

Question 10

An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the curve $y = 1 - \frac { x ^ { 2 } } { 4 } , - 2$ $\leq x \leq 2$, 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)  2 
B)  8 
C)  7 
D)  3 
A)  2 
B)  8 
C)  7 
D)  3

Question 11

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
-$$y = \sqrt { 2 x } , y = 2 , x = 0$$

Accepted Answer

A)  $2 \pi$ 
B)  $\frac { 8 } { 3 } \pi$ 
C)  $8 \pi$ 
D)  $4 \pi$ 
A)  $2 \pi$ 
B)  $\frac { 8 } { 3 } \pi$ 
C)  $8 \pi$ 
D)  $4 \pi$

Question 12

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
-$$y = x ^ { 2 } , y = 4 , x = 0$$

Accepted Answer

A)  $\frac { 32 } { 5 } \pi$ 
B)  $\frac { 128 } { 5 } \pi$ 
C)  $\frac { 192 } { 5 } \pi$ 
D)  $\frac { 16 } { 3 } \pi$ 
A)  $\frac { 32 } { 5 } \pi$ 
B)  $\frac { 128 } { 5 } \pi$ 
C)  $\frac { 192 } { 5 } \pi$ 
D)  $\frac { 16 } { 3 } \pi$

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.
-$$y = \frac { 4 } { \sqrt { x } } , y = 0 , x = 1 , x = 16$$

Accepted Answer

A)  $336 \pi$ 
B)  $168 \pi$ 
C)  $\frac { 1024 } { 3 } \pi$ 
D)  $\frac { 1040 } { 3 } \pi$ 
A)  $336 \pi$ 
B)  $168 \pi$ 
C)  $\frac { 1024 } { 3 } \pi$ 
D)  $\frac { 1040 } { 3 } \pi$

Question 14

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
-$$y = x + 1 , y = 0 , x = - 1 , x = 6$$

Accepted Answer

A)  $\frac { 343 } { 3 } \pi$ 
B)  $\frac { 7 } { 2 } \pi$ 
C)  $49 \pi$ 
D)  $24 \pi$ 
A)  $\frac { 343 } { 3 } \pi$ 
B)  $\frac { 7 } { 2 } \pi$ 
C)  $49 \pi$ 
D)  $24 \pi$

Question 15

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
-$$y = \sqrt { \sin 7 x } , y = 1 , x = 0 \text { to } x = \frac { \pi } { 14 }$$

Accepted Answer

A)  $\frac { \pi ^ { 2 } } { 14 } + \pi$ 
B)  $\frac { \pi } { 14 } - \frac { 1 } { 7 }$ 
C)  $\frac { \pi ^ { 2 } } { 14 } - \pi$ 
D)  $\frac { \pi ^ { 2 } } { 14 } - \frac { \pi } { 7 }$ 
A)  $\frac { \pi ^ { 2 } } { 14 } + \pi$ 
B)  $\frac { \pi } { 14 } - \frac { 1 } { 7 }$ 
C)  $\frac { \pi ^ { 2 } } { 14 } - \pi$ 
D)  $\frac { \pi ^ { 2 } } { 14 } - \frac { \pi } { 7 }$

Question 16

The disk $( x - 4 ) ^ { 2 } + y ^ { 2 } \leq 1$ is revolved about the $y$-axis to generate a torus. Find its volume. (Hint: $\int _ { - 1 } ^ { 1 } \sqrt { 1 - y ^ { 2 } } d y = \frac { 1 } { 2 } \pi$, since it is the area of a semicircle of radius 1 .)

Accepted Answer

A)  $8 \pi ^ { 2 }$ 
B)  $2 \pi ^ { 2 }$ 
C)  $4 \pi ^ { 2 }$ 
D)  $4 \pi$ 
A)  $8 \pi ^ { 2 }$ 
B)  $2 \pi ^ { 2 }$ 
C)  $4 \pi ^ { 2 }$ 
D)  $4 \pi$

Question 17

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
-$$y = \frac { 1 } { \sqrt { x } } , y = 0 , x = 1 , x = 6$$

Accepted Answer

A)  $\frac { \pi } { 2 } ( \ln 6 )$ 
B)  $\sqrt { 6 } \pi$ 
C)  $\pi ( \ln 6 )$ 
D)  $\frac { \pi } { 6 }$ 
A)  $\frac { \pi } { 2 } ( \ln 6 )$ 
B)  $\sqrt { 6 } \pi$ 
C)  $\pi ( \ln 6 )$ 
D)  $\frac { \pi } { 6 }$

Question 18

Find the volume of the solid generated by revolving the region about the given line.
-The region in the first quadrant bounded above by the line $y = 5 x ^ { 3 }$, below by $x$-axis, and on the right by the line $x = 1$, about the line $y = - 1$

Accepted Answer

A)  $\frac { 45 } { 14 } \pi$ 
B)  $\frac { 85 } { 14 } \pi$ 
C)  $\frac { 32 } { 7 } \pi$ 
D)  $\frac { 85 } { 7 } \pi$ 
A)  $\frac { 45 } { 14 } \pi$ 
B)  $\frac { 85 } { 14 } \pi$ 
C)  $\frac { 32 } { 7 } \pi$ 
D)  $\frac { 85 } { 7 } \pi$

Question 19

A frustum of a right circular cone has a height of $10 \mathrm {~m}$, a base of radius $2 \mathrm {~m}$, and a top of radius $1 \mathrm {~m}$. Find its volume.

Accepted Answer

A)  $\frac { 7 } { 3 } \pi$ 
B)  $\frac { 70 } { 3 } \pi$ 
C)  $70 \pi$ 
D)  $7 \pi$ 
A)  $\frac { 7 } { 3 } \pi$ 
B)  $\frac { 70 } { 3 } \pi$ 
C)  $70 \pi$ 
D)  $7 \pi$

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.
-$$x = 50 - y ^ { 2 } , x = y ^ { 2 } , y = 0$$

Accepted Answer

A)  $1250 \pi$ 
B)  $\frac { 625 } { 2 } \pi$ 
C)  $\frac { 625 } { 4 } \pi$ 
D)  $625 \pi$ 
A)  $1250 \pi$ 
B)  $\frac { 625 } { 2 } \pi$ 
C)  $\frac { 625 } { 4 } \pi$ 
D)  $625 \pi$

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = x ^ { 2 } , y = 0 , x = 0 , x = 5$

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. - $y = x ^ { 2 } - 5 , y = 4 x , x = 0 \text {, for } x \geq 0$

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves about the given lines. - $y = 4 x , \quad y = 0 , \quad x = 3$ ; revolve about the $x$ -axis

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = \frac { 1 } { x } , y = 0 , x = 1 , x = 9$

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

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = 3 \cos \pi x , y = 0 , x = - 0.5 , x = 0.5$

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. - $x = 3 y ^ { 2 } , x = - 3 y , y = 3$

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. - $y = 3 x , y = 6 x , x = 3$

Find the volume of the solid generated by revolving the region about the given line. -The region in the first quadrant bounded above by the line $5 x + y = 10$ , below by the $x$ -axis, and on the left by the $y$ -axis, about the line $x = - 2$

An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the curve $y = 1 - \frac { x ^ { 2 } } { 4 } , - 2$ $\leq x \leq 2$ , 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 volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = \sqrt { 2 x } , y = 2 , x = 0$

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = x ^ { 2 } , y = 4 , x = 0$

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. - $y = \frac { 4 } { \sqrt { x } } , y = 0 , x = 1 , x = 16$

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = x + 1 , y = 0 , x = - 1 , x = 6$

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = \sqrt { \sin 7 x } , y = 1 , x = 0 \text { to } x = \frac { \pi } { 14 }$

The disk $( x - 4 ) ^ { 2 } + y ^ { 2 } \leq 1$ is revolved about the $y$ -axis to generate a torus. Find its volume. (Hint: $\int _ { - 1 } ^ { 1 } \sqrt { 1 - y ^ { 2 } } d y = \frac { 1 } { 2 } \pi$ , since it is the area of a semicircle of radius 1 .)

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = \frac { 1 } { \sqrt { x } } , y = 0 , x = 1 , x = 6$

Find the volume of the solid generated by revolving the region about the given line. -The region in the first quadrant bounded above by the line $y = 5 x ^ { 3 }$ , below by $x$ -axis, and on the right by the line $x = 1$ , about the line $y = - 1$

A frustum of a right circular cone has a height of $10 \mathrm {~m}$ , a base of radius $2 \mathrm {~m}$ , and a top of radius $1 \mathrm {~m}$ . Find its volume.

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. - $x = 50 - y ^ { 2 } , x = y ^ { 2 } , y = 0$

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Exam 6: Applications of Definite Integrals

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - y=x2,y=0,x=0,x=5y = x ^ { 2 } , y = 0 , x = 0 , x = 5y=x2,y=0,x=0,x=5

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=x2−5,y=4x,x=0, for x≥0y = x ^ { 2 } - 5 , y = 4 x , x = 0 \text {, for } x \geq 0y=x2−5,y=4x,x=0, for x≥0

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves about the given lines. - y=4x,y=0,x=3y = 4 x , \quad y = 0 , \quad x = 3y=4x,y=0,x=3 ; revolve about the xxx -axis

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - y=1x,y=0,x=1,x=9y = \frac { 1 } { x } , y = 0 , x = 1 , x = 9y=x1​,y=0,x=1,x=9

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the xxx -axis y=2sin⁡xy = 2 \sqrt { \sin x }y=2sinx​

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - y=3cos⁡πx,y=0,x=−0.5,x=0.5y = 3 \cos \pi x , y = 0 , x = - 0.5 , x = 0.5y=3cosπx,y=0,x=−0.5,x=0.5

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=3y2,x=−3y,y=3x = 3 y ^ { 2 } , x = - 3 y , y = 3x=3y2,x=−3y,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=3x,y=6x,x=3y = 3 x , y = 6 x , x = 3y=3x,y=6x,x=3

Find the volume of the solid generated by revolving the region about the given line. -The region in the first quadrant bounded above by the line 5x+y=105 x + y = 105x+y=10 , below by the xxx -axis, and on the left by the yyy -axis, about the line x=−2x = - 2x=−2

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - y=2x,y=2,x=0y = \sqrt { 2 x } , y = 2 , x = 0y=2x​,y=2,x=0

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - y=x2,y=4,x=0y = x ^ { 2 } , y = 4 , x = 0y=x2,y=4,x=0

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=4x,y=0,x=1,x=16y = \frac { 4 } { \sqrt { x } } , y = 0 , x = 1 , x = 16y=x​4​,y=0,x=1,x=16

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - y=x+1,y=0,x=−1,x=6y = x + 1 , y = 0 , x = - 1 , x = 6y=x+1,y=0,x=−1,x=6

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - y=sin⁡7x,y=1,x=0 to x=π14y = \sqrt { \sin 7 x } , y = 1 , x = 0 \text { to } x = \frac { \pi } { 14 }y=sin7x​,y=1,x=0 to x=14π​

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - y=1x,y=0,x=1,x=6y = \frac { 1 } { \sqrt { x } } , y = 0 , x = 1 , x = 6y=x​1​,y=0,x=1,x=6

Find the volume of the solid generated by revolving the region about the given line. -The region in the first quadrant bounded above by the line y=5x3y = 5 x ^ { 3 }y=5x3 , below by xxx -axis, and on the right by the line x=1x = 1x=1 , about the line y=−1y = - 1y=−1

A frustum of a right circular cone has a height of 10 m10 \mathrm {~m}10 m , a base of radius 2 m2 \mathrm {~m}2 m , and a top of radius 1 m1 \mathrm {~m}1 m . Find its volume.

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=50−y2,x=y2,y=0x = 50 - y ^ { 2 } , x = y ^ { 2 } , y = 0x=50−y2,x=y2,y=0