Volumes of Solids of Revolution
Apply the disk and washer methods to compute volumes of solids formed by rotating regions about horizontal and vertical axes. Set up the integral carefully and integrate using FTC Part 2.
A lesson of AP Calculus AB, in Applications of Integration, about 10 minutes.
Volumes of Revolution: Disk, Washer, and Shell Methods
By the end of this lesson, you will be able to set up and evaluate integrals to find volumes of solids formed by revolving regions around horizontal and vertical axes.
The Concept
When a 2D region is revolved around an axis, it sweeps out a 3D solid. We can compute the volume by imagining slicing the solid into thin slices and integrating.
Disk Method (No Hole)
Revolving a region bounded by $y = f(x)$ and the x-axis around the x-axis:
$$V = \pi \int_a^b [f(x)]^2, dx$$
Each thin disk has radius $R = f(x)$ and thickness $dx$; its volume is $\pi R^2, dx$.
Example: Revolve $y = \sqrt{x}$ on $[0, 4]$ around the x-axis.
$V = \pi\int_0^4 (\sqrt{x})^2, dx = \pi\int_0^4 x, dx = \pi\left[\frac{x^2}{2}\right]_0^4 = \pi(8) = 8\pi$
Washer Method (With Hole)
Revolving the region between $y = f(x)$ (outer) and $y = g(x)$ (inner) around the x-axis:
$$V = \pi \int_a^b \left{[f(x)]^2 - [g(x)]^2\right}, dx$$
Each washer has outer radius $R = f(x)$ and inner radius $r = g(x)$.
Example: Revolve the region between $y = x$ (top) and $y = x^2$ (bottom) on $[0, 1]$ around the x-axis.
$V = \pi\int_0^1 [(x)^2 - (x^2)^2], dx = \pi\int_0^1 (x^2 - x^4), dx = \pi\left[\frac{x^3}{3} - \frac{x^5}{5}\right]_0^1 = \pi\left(\frac{1}{3}-\frac{1}{5}\right) = \frac{2\pi}{15}$
Revolving Around the y-axis
When revolving around the y-axis (or $x = 0$), use $x$ as a function of $y$:
$$V = \pi\int_c^d [h(y)]^2, dy$$
Alternative: the Shell Method is often easier for revolving around vertical axes.
Shell Method (Revolving Around y-axis)
$$V = 2\pi \int_a^b x \cdot f(x), dx$$
Each thin cylindrical shell at position $x$ has radius $x$, height $f(x)$, and thickness $dx$. The volume of a shell is $2\pi \cdot \text{radius} \cdot \text{height} \cdot \text{thickness} = 2\pi x f(x), dx$.
When to use shells: when the washer method requires solving for x as a function of y (which can be messy).
Example: Revolve the region under $y = x(2-x)$ (from 0 to 2) around the y-axis.