Answer
$\displaystyle{V=\frac{11\pi }{30}}\\ $
Work Step by Step
$\displaystyle{x^2=x^{\frac{1}{2}}}\\ \displaystyle{x^2-x^{\frac{1}{2}}=0}\\ \displaystyle{x^{\frac{1}{2}}\left(x^{\frac{1}{2}}-1\right)=0}\\ x=0 \qquad x=1$
$\displaystyle{A\left(x\right)=\pi \left(1-x^2\right)^2-\pi \left(1-x^{\frac{1}{2}}\right)^2}\\ \displaystyle{A\left(x\right)=\pi \left(x^4+2x^{\frac{1}{2}}-2x^2-x\right)}\\$
$\displaystyle{V=\int_{0}^1A\left(x\right)\ dx}\\ \displaystyle{V=\int_{0}^1\pi \left(x^4+2x^{\frac{1}{2}}-2x^2-x\right)\ dx}\\ \displaystyle{V=\pi \int_{0}^1x^4+2x^{\frac{1}{2}}-2x^2-x\ dx}\\ \displaystyle{V=\pi\left[\frac{1}{5}x^5-\frac{4}{3}x^{\frac{3}{2}}-\frac{2}{3}x^3-\frac{1}{2}x^2\right]_{0}^1}\\ \displaystyle{V=\pi\left(\left(\frac{1}{5}1^5-\frac{4}{3}1^{\frac{3}{2}}-\frac{2}{3}1^3-\frac{1}{2}1^2\right)-\left(\frac{1}{5}0^5-\frac{4}{3}0^{\frac{3}{2}}-\frac{2}{3}0^3-\frac{1}{2}0^2\right)\right)}\\ \displaystyle{V=\frac{11\pi }{30}}\\ $
