Find the volume of the solid obtained when the region bounded by the curves $y = x ^ { 2 } + 1 , y = 1 , \text { and } x = 1$ is rotated about the line x =1.
A) $\frac { \pi } { 12 }$
B) $8 \pi$
C) $\frac { \pi } { 2 }$
D) $\frac { \pi } { 4 }$
E) $\frac { \pi } { 3 }$
F) $\frac { \pi } { 6 }$
G) $4 \pi$
H) $\frac { 10 \pi } { 3 }$

Question

Find the volume of the solid obtained when the region bounded by the curves $y = x ^ { 2 } + 1 , y = 1 , \text { and } x = 1$ is rotated about the line x =1.
A) $\frac { \pi } { 12 }$ 
B) $8 \pi$ 
C) $\frac { \pi } { 2 }$ 
D) $\frac { \pi } { 4 }$ 
E) $\frac { \pi } { 3 }$ 
F) $\frac { \pi } { 6 }$ 
G) $4 \pi$ 
H) $\frac { 10 \pi } { 3 }$

Quizplus · Accepted Answer

The volume is calculated using the method of cylindrical shells. The height of the shell is $x^2$ and the radius is $1-x$. Integrating from 0 to 1 gives $\int_0^1 2\pi(1-x)(x^2) \, dx = \frac{\pi}{6}$.

Find the Volume of the Solid Obtained When the Region y=x2+1,y=1, and x=1y = x ^ { 2 } + 1 , y = 1 , \text { and } x = 1y=x2+1,y=1, and x=1

Find the Volume of the Solid Obtained When the Region $y = x ^ { 2 } + 1 , y = 1 , \text { and } x = 1$