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 $
$\bf{Calculations:}$
Here, we have $f(y)=-x^2+2x+2$ and $g(y)=2x^2-4x+2$
Thus, the total volume can be computed as:
$\text{Volume}, V=\pi \int_0^2 [(-x^2+2x+2)^2-(2x^2-4x+2)] \ dx$
or, $=\pi \int_0^2 (24x-24x^2+12x^3 -3x^4) \ dx$
or, $=\pi [\dfrac{24x^2}{2}-\dfrac{24x^3}{3}+\dfrac{12x^4}{4} -\dfrac{3x^5}{5}]_0^2$
or, $=\dfrac{64 \pi}{5}$
