Answer
$\displaystyle{V=4-\frac{8}{\pi}}\\$
Work Step by Step
$\displaystyle{V=\int_{0}^{1}(2\pi x)\left(\cos\left(\frac{\pi}{2}x\right)\right)dx}\\
\displaystyle{V=2\pi\int_{0}^{1} x\cos\left(\frac{\pi}{2}x\right)\ dx}\\$
$\displaystyle \left[\begin{array}{ll} u=x & dv=\cos\left(\frac{\pi}{2}x\right) \\ & \\ du=1 & v=\frac{2}{\pi}\sin\left(\frac{\pi}{2}x\right) \end{array}\right]$ Integration by parts
$\displaystyle{V=2\pi\left[\frac{2}{\pi}x\sin\left(\frac{\pi}{2}x\right)\right]_{0}^{1}-2\pi\int_{0}^{1}\frac{2}{\pi}\sin\left(\frac{\pi}{2}x\right)\ dx}\\
\displaystyle{V=2\pi\left(\frac{2}{\pi}\right)-4\int_{0}^{1}\sin\left(\frac{\pi}{2}x\right)\ dx}\\
\displaystyle{V=4-\left[-\frac{2}{\pi}\cos\left(\frac{\pi}{2}x\right)\right]_{0}^{1}}\\
\displaystyle{V=4-\frac{8}{\pi}}\\$
