Find the volume of the solid generated by revolving the region about the y-axis.
-The region enclosed by $x = y 1 / 3 , x = 0 , y = 27$
A) $\frac { 243 } { 4 } \pi$
B) $81 \pi$
C) $\frac { 729 } { 5 } \pi$
D) $243 \pi$

Question

Quizplus · Accepted Answer

The volume of the solid of revolution about the y-axis is calculated using the formula $ V = \pi \int_{a}^{b} [f(y)]^2 \, dy $. Here, $ f(y) = y^{1/3} $, and the limits are from $ y = 0 $ to $ y = 27 $. The integral becomes $ \pi \int_{0}^{27} (y^{1/3})^2 \, dy = \pi \int_{0}^{27} y^{2/3} \, dy $. Evaluating this integral gives $\frac{729}{5} \pi$.

Find the Volume of the Solid Generated by Revolving the Region