Answer
the volume of the solid is given by :
$$\begin{aligned}
V &=\int_{-1}^{1} A(x) d x \\
&= \int_{-1}^{1}\left[\left(2-x^{2}\right)-x^{2}\right]^{2} d x \\
&=\frac{64}{15}
\end{aligned}$$
Work Step by Step
the volume of the solid is given by :
$$\begin{aligned}
V &=\int_{-1}^{1} A(x) d x \\
&=2 \int_{0}^{1} A(x) d x \,\,\,\,\,\, \text {(The integrand is even.)}\\
&=2 \int_{0}^{1}\left[\left(2-x^{2}\right)-x^{2}\right]^{2} d x \\
&=2 \int_{0}^{1}\left[2\left(1-x^{2}\right)\right]^{2} d x \\
&=8 \int_{0}^{1}\left(1-2 x^{2}+x^{4}\right) d x \\
&=8\left[x-\frac{2}{3} x^{3}+\frac{1}{5} x^{5}\right]_{0}^{1} \\
&=8\left(1-\frac{2}{3}+\frac{1}{5}\right) \\
&=\frac{64}{15}
\end{aligned}$$
