Answer
$\displaystyle{V=\frac{\pi}{3}}$
Work Step by Step
$\displaystyle{A(x)=\pi\left(x^{\frac{1}{4}}\right)^2-\pi\left(x\right)^2}\\
\displaystyle{A(x)=\pi\left(x^{\frac{1}{2}}-x^2\right)}$
$\begin{aligned} V &=\int_{0}^{1} A(x) \ d x \\ V &=\int_{0}^{1} \pi\left(x^{\frac{1}{2}}-x^2\right) \ d x \\ V &=\pi \int_{0}^{1} x^{\frac{1}{2}}-x^2\ dx \\ V &=\pi\left[\frac{2}{3} x^{\frac{3}{2}}-\frac{1}{3} x^3\right]_{0}^{1} \\ V &=\pi\left(\left(\frac{2}{3} (1)^{\frac{3}{2}}-\frac{1}{3} (1)^3\right)-(0)\right) \\ V &=\frac{\pi}{3} \end{aligned}$
