Answer
$\dfrac{128 \pi}{5}$
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 $
$\bf{Calculations:}$
$V= \pi \int_0^2 [(4)^2-(y^{2})^2] \ dy$
or, $= \pi [16y -\dfrac{y^5}{5}]_0^2$
or, $=\pi [16(2) -\dfrac{(2)^5}{5}]$
or, $=32 \pi- \dfrac{32 \pi }{5}$
Hence, $V=\dfrac{128 \pi}{5}$
