## Calculus 10th Edition

$\frac{\pi}{2}$
Setup the integration using shell method about the x-axis $2\pi \int_{\frac{1}{2}}^1 y(\frac{1}{y} -1)dy + 2\pi \int_0^{\frac{1}{2}} ydy$ $2\pi \int_{\frac{1}{2}}^1 (1-y)dy + 2\pi \int_0^{\frac{1}{2}}y dy$ , Integrate $2\pi(y-\frac{1}{2}y^2)]^1_{\frac{1}{2}} + \pi(y^2)]_0^{\frac{1}{2}}$, Take definite integral $2\pi ((1-\frac{1}{2})-(\frac{3}{8})) + \pi(\frac{1}{4})$, Simplify $\frac{\pi}{4} + \frac{\pi}{4}$ $\frac{\pi}{2}$