Answer
$\displaystyle{V=\frac{3\pi }{5}}\\ $
Work Step by Step
$\displaystyle{A\left(y\right)=\pi \left(2-\sqrt[3] y\right)^2-\pi \left(2-1\right)^2}\\
\displaystyle{A\left(y\right)=\pi \left(3+y^{\frac{2}{3}}-4y^{\frac{1}{3}}\right)}\\$
$\displaystyle{V=\int_0^1A\left(y\right)\ dy}\\
\displaystyle{V=\int_0^1\pi \left(3+y^{\frac{2}{3}}-4y^{\frac{1}{3}}\right)\ dy}\\
\displaystyle{V=\pi \int_0^13+y^{\frac{2}{3}}-4y^{\frac{1}{3}}\ dy}\\
\displaystyle{V=\pi\left[3y+\frac{3}{5}y^{\frac{5}{3}}-3y^{\frac{4}{3}}\right]_0^1}\\
\displaystyle{V=\pi\left(\left(3(1)+\frac{3}{5}(1)^{\frac{5}{3}}-3(1)^{\frac{4}{3}}\right)-\left(0\right)\right)}\\
\displaystyle{V=\frac{3\pi }{5}}\\ $
