Answer
$\dfrac{64 \pi}{5}$
Work Step by Step
Our aim is to compute the volume of the revolution of the curve about the y-axis by using the Washer method.
Washer method for computing the volume of the revolution of the curve: Let us consider two functions $f(x)$ and $g(x)$ (both are continuous functions) with $f(x) \geq g(x) \geq 0$ on the interval $[m, n]$ . Then the volume of the solid can be obtained by rotating the region under the graph about the y-axis and can be expressed as: $\ Volume, V=\pi \int_m^n [f(y)^2-g(y)^2] \ dy $
$\bf{Calculations:}$
$V= \pi \int_0^8 [(2)^2-(y^{1/3})^2] \ dy$
or, $= \pi[4y-\dfrac{y^{5/3}}{5/3}]_0^8$
or, $=(4 \pi)\times 8- \dfrac{8^{5/3} \pi}{5/3}$
or, $=32 \pi- \dfrac{8^{5/3} \pi}{5/3}$
Hence, $V=\dfrac{64 \pi}{5}$
