Volumes of Revolution
Volumes of Revolution
A thin strip perpendicular to the axis of rotation generates a disk. Its cross-sectional area is \(\pi r^2\), where \(r\) is the distance from the strip to the axis.
Rotation about the x-axis
Disk method
\[V=\pi\int_a^b[\mathrm{f}(x)]^2\,\mathrm{d}x\]
Rotation about the y-axis
Disk method
\[V=\pi\int_c^d[\mathrm{f}(y)]^2\,\mathrm{d}y\]
Use radii, not signed coordinates: squaring produces the disk area. Limits must match the variable of integration.
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Answer:About the \(x\)-axis: \(V=\pi\int_a^b y^2\,\mathrm dx\) About the \(y\)-axis: \(V=\pi\int_a^b x^2\,\mathrm dy\)
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