Question 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 y-axis.
-$$y = 8 - x ^ { 2 } , y = x ^ { 2 } , x = 0$$

Accepted Answer

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

Question 2

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

Accepted Answer

A)  $2 \pi$ 
B)  $\frac { 16 } { 3 } \pi$ 
C)  $8 \pi$ 
D)  $4 \pi$

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 $x$-axis

Accepted Answer

A)  $18 \pi$ 
B)  $27 \pi$ 
C)  $\frac { 9 } { 2 } \pi$ 
D)  $9 \pi$

Question 4

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.
-$$y = 4 | x | , y = 4$$

Accepted Answer

A)  $\frac { 32 } { 3 } \pi$ 
B)  $\frac { 4 } { 3 } \pi$ 
C)  $\frac { 64 } { 3 } \pi$ 
D)  $\frac { 1024 } { 3 } \pi$

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 $y = 1$

Accepted Answer

A)  $\frac { 10 } { 7 } \pi$ 
B)  $\frac { 15 } { 7 } \pi$ 
C)  $\frac { 20 } { 7 } \pi$ 
D)  $\frac { 15 } { 14 } \pi$

Question 6

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.
-$y = 2 x , \quad y = 0 , \quad x = 3$; revolve about the line $x = - 4$

Accepted Answer

A)  $54 \pi$ 
B)  $108 \pi$ 
C)  $- 36 \pi$ 
D)  54

Question 7

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

Accepted Answer

A)  $27 \pi$ 
B)  $\frac { 45 } { 2 } \pi$ 
C)  $18 \pi$ 
D)  $\frac { 45 } { 4 } \pi$

Question 8

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

Accepted Answer

A)  $1250 \pi$ 
B)  $50 \pi$ 
C)  $\frac { 50 } { 3 } \pi$ 
D)  $25 \pi$

Question 9

A right circular cylinder is obtained by revolving the region enclosed by the line $x = r$, the $x$-axis, and the line $y = h$, about the $y$-axis. Find the volume of the cylinder.

Accepted Answer

A)  $\pi \mathrm { rh }$ 
B)  $2 \pi r ^ { 2 } \mathrm {~h}$ 
C)  $\pi \mathrm { rh } ^ { 2 }$ 
D)  $\pi r ^ { 2 } h$

Question 10

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 $y = 6$, below by the line $y = \frac { 6 x } { 5 }$, and on the left by the $y$-axis, about the line $y = 6$

Accepted Answer

A)  $30 \pi$ 
B)  $50 \pi$ 
C)  $60 \pi$ 
D)  $420 \pi$

Question 11

Find the volume of the solid generated by revolving the shaded region about the given axis.
-About the $y$-axis
 $y = \sqrt { 3 x }$

Accepted Answer

A)  $18 \pi$ 
B)  $\frac { 243 } { 5 } \pi$ 
C)  $3 \pi$ 
D)  $\frac { 27 } { 5 } \pi$

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 y-axis.
-$$y = 4 x , y = - \frac { x } { 4 } , x = 1$$

Accepted Answer

A)  $\frac { 17 } { 6 } \pi$ 
B)  $\frac { 17 } { 12 } \pi$ 
C)  $\frac { 11 } { 4 } \pi$ 
D)  $17 \pi$

Question 13

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 $y = 5$, below by the curve $y = \sqrt { 5 x }$, and on the left by the $y$-axis, about the line $y = 5$

Accepted Answer

A)  $\frac { 125 } { 6 } \pi$ 
B)  $\frac { 25 } { 3 } \pi$ 
C)  $\frac { 125 } { 2 } \pi$ 
D)  $25 \pi$

Question 14

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.
-$y = 4 x , \quad y = x ^ { 2 }$; revolve about the $y$-axis

Accepted Answer

A)  $\frac { 64 } { 3 } \pi$ 
B)  $\frac { 896 } { 3 } \pi$ 
C)  $- \frac { 128 } { 3 } \pi$ 
D)  $\frac { 128 } { 3 } \pi$

Question 15

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
-$$y = 7 \cos ( \pi x ) , y = 7 , x = - 0.5 , x = 0.5$$

Accepted Answer

A)  $98 \pi$ 
B)  $49 \pi$ 
C)  $\frac { 49 } { 3 } \pi$ 
D)  $\frac { 49 } { 2 } \pi$

Question 16

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.
-$$y = \frac { 5 } { x } , y = 0 , x = 5 , x = 7$$

Accepted Answer

A)  $15 \pi$ 
B)  $30 \pi$ 
C)  $10 \pi$ 
D)  $20 \pi$

Question 17

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
-$$y = x , y = 0 , x = 1 , x = 5$$

Accepted Answer

A)  $\frac { 124 } { 3 } \pi$ 
B)  $\frac { 4 } { 3 } \pi$ 
C)  $13 \pi$ 
D)  $12 \pi$

Question 18

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

$y = 4 - x ^ { 2 }$

Accepted Answer

A)  $\frac { 16 } { 3 } \pi$ 
B)  $\frac { 416 } { 5 } \pi$ 
C)  $\frac { 416 } { 15 } \pi$ 
D)  $\frac { 256 } { 15 } \pi$

Question 19

Find the volume of the solid generated by revolving the shaded region about the given axis.
-About the $y$-axis
 $x = 2 \tan \left( \frac { y } { 5 } \right)$

Accepted Answer

A)  $20 \pi$ 
B)  $10 \pi ^ { 2 } + 5 \pi$ 
C)  $10 \pi ^ { 2 } - 20 \pi$ 
D)  $5 \pi ^ { 2 } - 10 \pi$

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.
-$$y = \sqrt { x } , y = 0 , y = x - 6$$

Accepted Answer

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

Exam 6: Applications of Definite Integrals

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 bounded by the given lines and curves about the x-axis. - $y = \sqrt { x } , y = 0 , x = 0 , x = 4$

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. - $y = 4 | x | , y = 4$

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. - $y = 2 x , \quad y = 0 , \quad x = 3$ ; revolve about the line $x = - 4$

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

A right circular cylinder is obtained by revolving the region enclosed by the line $x = r$ , the $x$ -axis, and the line $y = h$ , about the $y$ -axis. Find the volume of the cylinder.

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 $y = 6$ , below by the line $y = \frac { 6 x } { 5 }$ , and on the left by the $y$ -axis, about the line $y = 6$

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

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 $y = 5$ , below by the curve $y = \sqrt { 5 x }$ , and on the left by the $y$ -axis, about the line $y = 5$

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. - $y = 4 x , \quad y = x ^ { 2 }$ ; revolve about the $y$ -axis

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = 7 \cos ( \pi x ) , y = 7 , x = - 0.5 , x = 0.5$

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 y-axis. - $y = \frac { 5 } { x } , y = 0 , x = 5 , x = 7$

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = x , y = 0 , x = 1 , x = 5$

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 x-axis. - $y = \sqrt { x } , y = 0 , y = x - 6$

Exam 6: Applications of Definite Integrals

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−x2,y=x2,x=0y = 8 - x ^ { 2 } , y = x ^ { 2 } , x = 0y=8−x2,y=x2,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,y=0,x=0,x=4y = \sqrt { x } , y = 0 , x = 0 , x = 4y=x​,y=0,x=0,x=4

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. -About the xxx -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 x-axis. - y=4∣x∣,y=4y = 4 | x | , y = 4y=4∣x∣,y=4

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=1y = 1y=1

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=2x,y=0,x=3y = 2 x , \quad y = 0 , \quad x = 3y=2x,y=0,x=3 ; revolve about the line x=−4x = - 4x=−4

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

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by x=y25,x=0,y=−5,y=5x = \frac { y ^ { 2 } } { 5 } , x = 0 , y = - 5 , y = 5x=5y2​,x=0,y=−5,y=5

A right circular cylinder is obtained by revolving the region enclosed by the line x=rx = rx=r , the xxx -axis, and the line y=hy = hy=h , about the yyy -axis. Find the volume of the cylinder.

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=6y = 6y=6 , below by the line y=6x5y = \frac { 6 x } { 5 }y=56x​ , and on the left by the yyy -axis, about the line y=6y = 6y=6

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the yyy -axis y=3xy = \sqrt { 3 x }y=3x​

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=−x4,x=1y = 4 x , y = - \frac { x } { 4 } , x = 1y=4x,y=−4x​,x=1

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=5y = 5y=5 , below by the curve y=5xy = \sqrt { 5 x }y=5x​ , and on the left by the yyy -axis, about the line y=5y = 5y=5

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=x2y = 4 x , \quad y = x ^ { 2 }y=4x,y=x2 ; revolve about the yyy -axis

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - y=7cos⁡(πx) ,y=7,x=−0.5,x=0.5y = 7 \cos ( \pi x ) , y = 7 , x = - 0.5 , x = 0.5y=7cos(πx) ,y=7,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 y-axis. - y=5x,y=0,x=5,x=7y = \frac { 5 } { x } , y = 0 , x = 5 , x = 7y=x5​,y=0,x=5,x=7

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - y=x,y=0,x=1,x=5y = x , y = 0 , x = 1 , x = 5y=x,y=0,x=1,x=5

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the xxx -axis y=4−x2y = 4 - x ^ { 2 }y=4−x2

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the yyy -axis x=2tan⁡(y5) x = 2 \tan \left( \frac { y } { 5 } \right) x=2tan(5y​)

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=x,y=0,y=x−6y = \sqrt { x } , y = 0 , y = x - 6y=x​,y=0,y=x−6