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Compute the volume of the solid of revolution generated by f(x) = sin x

by RoRi

Sketch the graph and compute the volume of the solid of revolution generated by:

    \[ f(x) = \sin x, \qquad \text{on} \qquad 0 \leq x \leq \pi. \]

The sketch of the ordinate set of f(x) = \sin x on [0,\pi] is as follows:

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We then compute the volume of the solid of revolution.

    \begin{align*} V = \int_0^{\pi} \pi \sin^2 x \, dx &= \frac{\pi}{2} \int_0^{\pi} (1 - \cos (2x)) \, dx \\ &= \frac{\pi}{2} \pi \\ &= \frac{\pi^2}{2}. \end{align*}

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