# Lecture 23: Applications — Volumes of Revolution

## Calculus I — Differential Calculus

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## Learning Objectives

By the end of this lecture, students will be able to:

1. Visualize solids of revolution formed by rotating a region about an axis
2. Compute volumes using the disk method
3. Compute volumes using the washer method
4. Compute volumes using the cylindrical shell method
5. Choose the appropriate method based on the axis of rotation and the given curves

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## Lecture Content

### I. Solids of Revolution

A **solid of revolution** is formed by rotating a plane region about a line called the axis of revolution. Common axes include the x-axis, the y-axis, and other horizontal or vertical lines. The volume of such solids can be computed using integration, and three main methods are available: the disk method, the washer method, and the shell method.

### II. The Disk Method

The disk method is used when the cross-section perpendicular to the axis of revolution is a solid disk with no hole. When **rotating about the x-axis**, the region between y = f(x) and the x-axis on [a, b] generates a solid whose cross-section at position x is a disk with radius r = f(x). The volume of a thin disk is pi * [f(x)]^2 * dx, so the total volume is V = integral from a to b of pi * [f(x)]^2 dx. When **rotating about the y-axis**, with the boundary given as x = g(y), the formula becomes V = integral from c to d of pi * [g(y)]^2 dy.

For example, rotating y = sqrt(x) about the x-axis from x = 0 to x = 4 gives V = integral from 0 to 4 of pi * (sqrt(x))^2 dx = pi * integral from 0 to 4 of x dx = pi * [x^2/2] from 0 to 4 = 8*pi.

<image>A three-part illustration of the disk method. Panel A: The region under y = sqrt(x) from x = 0 to 4 shown in 2D. Panel B: The same region being rotated about the x-axis, with a few representative disks drawn at different x-values, showing the 3D solid forming. Panel C: A single representative disk at position x, with radius r = sqrt(x) and thickness dx labeled. The formula V = pi * integral of [f(x)]^2 dx is displayed. Title: "The Disk Method: rotating about the x-axis."</image>

### III. The Washer Method

The washer method is used when the cross-section is an annulus (a disk with a hole), which occurs when the region lies between two curves. When **rotating about the x-axis**, with y = f(x) as the outer curve and y = g(x) as the inner curve on [a, b], the outer radius is R = f(x) and the inner radius is r = g(x). The volume is V = integral from a to b of pi * ([f(x)]^2 - [g(x)]^2) dx.

For instance, rotating the region between y = x and y = x^2 about the x-axis on [0, 1] (where x >= x^2, so x is the outer radius) gives V = pi * integral from 0 to 1 of (x^2 - x^4) dx = pi * [x^3/3 - x^5/5] from 0 to 1 = pi(1/3 - 1/5) = 2*pi/15.

When rotating about a line other than a coordinate axis, the radii must be adjusted to represent distances from the curves to the axis of rotation. For example, rotating the region under y = x^2 about y = 4 requires measuring the outer radius as R = 4 - 0 = 4 (from y = 4 to the x-axis) and the inner radius as r = 4 - x^2 (from y = 4 to the curve), or vice versa depending on the geometry.

<image>Panel A: A 2D region between y = x and y = x^2 on [0, 1]. Panel B: The 3D solid formed by rotating this region about the x-axis. A representative washer cross-section is shown with outer radius R = x and inner radius r = x^2, creating a ring shape. The washer is highlighted with the hole clearly visible. The formula V = pi * integral of (R^2 - r^2) dx is displayed. Title: "The Washer Method: the region has a hole."</image>

### IV. The Shell Method

The shell method uses **cylindrical shells** instead of disks or washers. It is particularly useful when rotating about the y-axis but the function is given as y = f(x). When **rotating about the y-axis**, a thin vertical strip at position x with width dx and height f(x) is rotated to form a cylindrical shell with radius x, height f(x), and thickness dx. The volume of this thin shell is 2*pi * x * f(x) * dx, giving V = integral from a to b of 2*pi * x * f(x) dx (with a >= 0).

When **rotating about the x-axis** with the region described by x = g(y), the formula becomes V = integral from c to d of 2*pi * y * g(y) dy.

For example, rotating y = x^2 on [0, 2] about the y-axis gives V = 2*pi * integral from 0 to 2 of x * x^2 dx = 2*pi * integral from 0 to 2 of x^3 dx = 2*pi * [x^4/4] from 0 to 2 = 2*pi * 4 = 8*pi.

### V. Choosing the Right Method

The **disk/washer method** is best when the slicing is perpendicular to the axis of rotation and the functions are easily expressed in terms of the variable along the axis. It is the natural choice when rotating about the x-axis with y = f(x), or about the y-axis with x = g(y).

The **shell method** is best when the slicing is parallel to the axis of rotation, or when using disk/washer would require solving for the other variable or splitting into multiple integrals. It is particularly convenient when rotating about the y-axis with y = f(x) given.

A useful rule of thumb: if the representative strips you draw are parallel to the axis of rotation, use shells; if they are perpendicular, use disk/washer.

<image>A decision chart for choosing the volume method. Two columns: "Axis of Rotation" and "Method." Row 1: Rotating about x-axis, function y = f(x) — use Disk/Washer (perpendicular slices). Row 2: Rotating about y-axis, function y = f(x) — use Shell (parallel slices) or rewrite as x = g(y) and use Disk/Washer. Row 3: Rotating about y-axis, function x = g(y) — use Disk/Washer. Row 4: Rotating about x-axis, function x = g(y) — use Shell. Each row has a small diagram showing the slice orientation. Title: "Which volume method should I use?"</image>

### VI. Volumes with Rotation About Other Lines

When rotating about a vertical line x = k (not the y-axis), the shell method uses radius |x - k| and height f(x), while the washer method adjusts radii by measuring distances from the curves to x = k. When rotating about a horizontal line y = k (not the x-axis), the washer method uses R = |f(x) - k| and r = |g(x) - k|, while the shell method uses radius |y - k|.

For example, rotating y = x^2 for 0 <= x <= 1 about x = 2 with the shell method gives V = 2*pi * integral from 0 to 1 of (2 - x) * x^2 dx = 2*pi * integral from 0 to 1 of (2x^2 - x^3) dx = 2*pi * [2x^3/3 - x^4/4] from 0 to 1 = 2*pi * (2/3 - 1/4) = 2*pi * 5/12 = 5*pi/6.

### VII. Summary of Volume Formulas

The three volume formulas are: **Disk**: V = pi * integral of [R(x)]^2 dx (or dy); **Washer**: V = pi * integral of ([R(x)]^2 - [r(x)]^2) dx (or dy); and **Shell**: V = 2*pi * integral of (radius)(height) dx (or dy). All three methods yield the same answer for a given solid -- the choice is purely a matter of convenience. Before writing the integral, always sketch the region and the solid, identify the axis of rotation, and draw a representative slice or shell.