Find the volume of the solid generated by revolving the shaded region about the given axis.
-About the $y$-axis
$y = \sqrt { 3 x }$
A) $18 \pi$
B) $\frac { 243 } { 5 } \pi$
C) $3 \pi$
D) $\frac { 27 } { 5 } \pi$

Question

Find the volume of the solid generated by revolving the shaded region about the given axis.
-About the $y$-axis
 $y = \sqrt { 3 x }$
A) $18 \pi$ 
B) $\frac { 243 } { 5 } \pi$ 
C) $3 \pi$ 
D) $\frac { 27 } { 5 } \pi$

Quizplus · Accepted Answer

We know that the region between the graph of $y=\sqrt{3x}$ and the $x$-axis from $x=0$ to $x=3$ creates a triangular shape. When revolved around the $y$-axis, this shape will create a cone and we can find the volume of this cone using the formula $\frac{1}{3}\pi r^2h$. The radius ($r$) is the distance from the $y$-axis to the right side of the triangle, which is $3$. The height ($h$) is the distance from the $x$-axis to the top of the triangle, which is $\sqrt{3(3)}=3\sqrt{3}$. 
$$\frac{1}{3}\pi (3)^2(3\sqrt{3})=\frac{27}{5}\pi$$
Therefore, the volume of the solid generated by revolving the shaded region about the $y$-axis is $\frac{27}{5}\pi$, which is answer choice $\boxed{\textbf{(D)}}$.

Find the Volume of the Solid Generated by Revolving the Shaded