Find the volume of the solid obtained by rotating the region bounded by $y = x$ and $x = y ^ { 3 }$ about the $x$-axis.
Select the correct answer.
A) $16 \pi$
B) $\frac { 7 } { 2 }$
C) $\frac { 4 } { 15 } \pi$
D) $\frac { 18 } { 35 }$
E) $\frac { 16 } { 7 } \pi$

Question

Quizplus · Accepted Answer

To solve this problem using the disk method, we can integrate with respect to $x$. Since the region is bounded by $y=x$ and $x=y^3$, we can rewrite these equations as $x=y$ and $y=x^{1/3}$. Thus, the integral for the volume becomes $$\int_{0}^{1} \pi\left((y^2)-(x^{2/3})^2\right) dy.$$ Simplifying and integrating, we get $$\int_{0}^{1} \pi\left(y^2-x^{4/3}\right) dy = \pi\left(\frac{y^3}{3}-\frac{x^{7/3}}{7/3}\right)\bigg|_{0}^{1} = \boxed{\frac{4}{15}\pi}.$$

Find the Volume of the Solid Obtained by Rotating the Region