When the rotating region does not touch the axis, each perpendicular strip generates a washer. Subtr

Question

When the rotating region does not touch the axis, each perpendicular strip generates a washer. Subtract the inner disk from the outer disk.

Washers about the x-axis

Outer radius minus inner radius

\[V=\pi\int_a^b\left([\mathrm{g}(x)]^2-[\mathrm{f}(x)]^2\right)\,\mathrm{d}x\]

Washers about the y-axis

Cylinder minus piecewise inner volume

\[\begin{aligned}V={}&\pi R^2(c-a)-\pi\int_a^b[\mathrm{f}(y)]^2\,\mathrm{d}y\\&-\pi\int_b^c[\mathrm{g}(y)]^2\,\mathrm{d}y\end{aligned}\]

Equivalently, write each piece as $\pi\int(R_{\rm outer}^2-R_{\rm inner}^2)\,\mathrm{d}y$. Split at $y=b$ because the inner boundary changes there.

Accepted Answer

About the $x$-axis: $V=\pi\int_a^b(g^2-f^2)\,\mathrm dx$ About the $y$-axis in the diagram: $V=\pi r^2(c-a)-\pi\int_a^b f^2\,\mathrm dy-\pi\int_b^c g^2\,\mathrm dy$

Volumes Between Curves

Volumes Between Curves

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