Volumes by Disks, Washers, and Shells Cheat Sheet

A printable reference covering disk, washer, and shell volume formulas, radius setup, axis choice, and integration bounds for grades 11-12.

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Volumes by disks, washers, and shells help students find the volume of a solid formed by rotating a region around an axis. This cheat sheet explains when to use each method and how to build the correct integral. Students need it because most errors come from choosing the wrong radius, bounds, or variable. The goal is to turn a picture of a region into a clear volume formula.

Key Facts

The disk method is used when cross sections perpendicular to the axis of rotation are solid circles, with volume $V = \pi \int_a^b [R(x)]^2\,dx$ or $V = \pi \int_c^d [R(y)]^2\,dy$ .
The washer method is used when cross sections perpendicular to the axis of rotation have a hole, with volume $V = \pi \int_a^b \left([R(x)]^2 - [r(x)]^2\right)\,dx$ .
For washers, the outer radius $R$ is the distance from the axis of rotation to the farther curve, and the inner radius $r$ is the distance to the closer curve.
The shell method is used with slices parallel to the axis of rotation, with volume $V = 2\pi \int_a^b (\text{radius})(\text{height})\,dx$ or $V = 2\pi \int_c^d (\text{radius})(\text{height})\,dy$ .
A shell radius is the distance from the slice to the axis of rotation, such as $r(x) = x$ for rotation around the $y$ -axis or $r(x) = x - a$ for rotation around $x = a$ .
A shell height is the length of the slice through the region, often $h(x) = f(x) - g(x)$ for vertical slices or $h(y) = x_{\text{right}} - x_{\text{left}}$ for horizontal slices.
Use $dx$ when slices are vertical and bounds are $x$ -values, and use $dy$ when slices are horizontal and bounds are $y$ -values.
If the axis of rotation is not one of the coordinate axes, radii must be written as distances, such as $R(x) = 5 - f(x)$ or $r(y) = g(y) - 2$ .

Vocabulary

Disk Method: A volume method that uses circular cross sections with no hole, usually written as $V = \pi \int [R]^2\,d x$ or $V = \pi \int [R]^2\,d y$ .
Washer Method: A volume method that uses cross sections shaped like washers, with volume found by subtracting the inner circular area from the outer circular area.
Shell Method: A volume method that adds thin cylindrical shells using $V = 2\pi \int (\text{radius})(\text{height})\,d x$ or $V = 2\pi \int (\text{radius})(\text{height})\,d y$ .
Axis of Rotation: The line around which a two-dimensional region is rotated to create a three-dimensional solid.
Outer Radius: In the washer method, the outer radius is the greater distance from the axis of rotation to the boundary of the region.
Inner Radius: In the washer method, the inner radius is the smaller distance from the axis of rotation to the boundary of the hole.

Common Mistakes to Avoid

Using $R - r$ instead of $R^2 - r^2$ for washers is wrong because washer area comes from subtracting circle areas, so the correct expression is $\pi(R^2 - r^2)$ .
Forgetting that radii are distances is wrong because a radius cannot be negative, so expressions like $R(x)$ should measure positive distance from the axis of rotation.
Mixing $dx$ bounds with $y$ -based radii is wrong because the variable of integration must match the slice direction and the interval limits.
Choosing shells when the height is not written as top minus bottom or right minus left causes errors because shell height must represent the full length of the slice inside the region.
Ignoring an axis shift such as $x = 3$ or $y = -2$ is wrong because radii must be measured from that shifted line, not automatically from the $x$ -axis or $y$ -axis.

Practice Questions

1 Find the volume when the region under $y = x^2$ from $x = 0$ to $x = 2$ is rotated about the $x$ -axis using disks.
2 Set up and evaluate the washer integral for the region between $y = 4$ and $y = x^2$ rotated about the $x$ -axis.
3 Use cylindrical shells to find the volume when the region under $y = 6 - x$ from $x = 0$ to $x = 6$ is rotated about the $y$ -axis.
4 A region is easier to describe with vertical slices, but it is rotated around a vertical axis. Explain why the shell method may be simpler than washers.

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