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

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

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


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

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We then compute the volume of the solid of revolution. We reference the previous exercises here and here for the integrals of \sin^2 x and \cos^2 x.

    \begin{align*}     V = \int_0^{\pi} \pi (\sin x + \cos x)^2 \, dx &= \pi \left(\int_0^{\pi} \sin^2 x \, dx + \int_0^{\pi} 2 \sin x \cos x \, dx + \int_0^{\pi} \cos^2 x \, dx \right) \\  &= \pi \left( \frac{\pi}{2} + 0 + \frac{\pi}{2} \right) \\  &= \pi^2. \end{align*}

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