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Compute the volume of the solid of revolution generated by the region between two functions

by RoRi

Let

    \[ f(x) = \sqrt{4 - x^2}, \qquad g(x) = 1, \qquad 0 \leq x \leq \sqrt{3}.\]

Sketch the graph of region between these functions and compute the volume of the solid of revolution generated by revolving this region about the x-axis.

The sketch of the region between f and g is as follows:

Rendered by QuickLaTeX.com

We then compute the volume of the solid of revolution as follows.

    \begin{align*} V &= \pi \int_0^{\sqrt{3}} (4-x^2 - 1) \, dx \\ &= 3 \pi \int_0^{\sqrt{3}} dx - \pi \int_0^{\sqrt{3}} x^2 \, dx \\ &= 3 \sqrt{3} \pi - \sqrt{3} \pi \\ &= 2 \sqrt{3} \pi. \end{align*}

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