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

A) $19 \pi$
B) $18 \pi$
C) $\frac { 19 } { 3 } \pi$
D) $\frac { 38 } { 3 } \pi$

Question

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

A) $19 \pi$ 
B) $18 \pi$ 
C) $\frac { 19 } { 3 } \pi$ 
D) $\frac { 38 } { 3 } \pi$

Quizplus · Accepted Answer

We use the shell method to find the volume. The shells will be taken along the height of the rectangle. The height of the rectangle is $3-x$ and the length of the rectangle is $2$. The axis of revolution is the $y$-axis, so the radius of each shell is $x$. Thus, the volume of each shell is $2\pi x(3-x)$. Integrating with respect to $x$ from $x=1$ to $x=2$, we get $$V=\int_1^2 2\pi x(3-x)\ dx=\left.\pi\left(x^3-\frac{3}{2}x^2ight)ight|_1^2=\frac{38}{3}\pi.$$Therefore, the answer is $\boxed{	extbf{(D)}}$.

Use the Shell Method to Find the Volume of the Solid