Volumes and Surface Areas of Solids of Revolution: Parametric and Polar Methods

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What is a solid of revolution?

A three-dimensional solid formed by rotating a plane figure or region around a fixed straight line, called the axis of revolution.

What distinguishes a disc/washer slice from a shell slice?

A disc or washer comes from a slice perpendicular to the axis of revolution. A shell comes from a slice parallel to the axis.

What is the area of a washer?

For outer radius $R$ and inner radius $r$, its area is $\pi(R^2-r^2)$, the outer disk area minus the inner disk area.

What is the shell volume formula for a thin slice?

A shell of radius $r$, height $h$, and thickness $dx$ has approximate volume $dV=2\pi rh\,dx$.

State Pappus's second centroid theorem for volume.

If a plane region of area $A$ rotates around an axis in its plane that does not cross the region, the volume is the area times the centroid's circular path length: $V=2\pi dA$, where $d$ is the centroid's distance from the axis.

Quiz questions

1. What is the fixed straight line around which a plane region is rotated called?Answer: The axis of revolution
2. What cross-section is formed when a thin slice perpendicular to the axis of revolution is rotated?Answer: A disc or washer
3. What is the approximate volume element for a cylindrical shell with radius $r$, height $h$, and thickness $dx$?Answer: $2\pi rh\,dx$
4. A plane region of area $A$ has a centroid distance $d$ from an in-plane axis that does not cross the region. What volume does Pappus's second centroid theorem give?Answer: $2\pi dA$
5. For a polar curve $r=f(\theta)$, what is the arc-length speed with respect to $\theta$?Answer: $\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}$