Volume of the solid of revolution for $f(x)=\sin x$
Find the colume of the solid obtained by rotating the graph of the function $y=\sin x$ around $x$-axis between $x=0$ and $x=\pi$.
Answer
The volume is given by the following integral
\[V=\int_0^{\pi}\pi \sin ^2 x dx=\pi \int_0^{\pi}\frac{1-\cos 2x}{2}dx\]
\[=\pi \left( \frac{x}{2}-\frac{\sin 2x}{4} \right) \bigg |_{x=0}^{x=\pi}=\pi (\frac{\pi}{2}-0)\]
\[=\frac{\pi^2}{2}.\]
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