Volumes of Revolution (Disk, Washer, Shell) Cheat Sheet

Key Facts

• The disk method is used when cross sections perpendicular to the axis of rotation are solid circles, with volume $V = \pi \int_a^b [R(x)]^2\,dx$.
• The washer method is used when cross sections perpendicular to the axis of rotation have holes, with volume $V = \pi \int_a^b \left([R(x)]^2 - [r(x)]^2\right)\,dx$.
• For rotation around the $x$-axis using vertical slices, the radius is usually a vertical distance such as $R(x) = y_{\text{top}} - 0$.
• For rotation around the $y$-axis using horizontal slices, the radius is usually a horizontal distance such as $R(y) = x_{\text{right}} - 0$.
• The shell method uses slices parallel to the axis of rotation, with volume $V = 2\pi \int_a^b p(x)h(x)\,dx$ or $V = 2\pi \int_c^d p(y)h(y)\,dy$.
• In the shell formula, $p$ is the distance from the slice to the axis of rotation and $h$ is the length of the slice across the region.
• When rotating around a line such as $y = k$ or $x = k$, radii must be distances to that line, such as $R(x) = |f(x) - k|$.
• Bounds of integration must match the variable of integration, so $dx$ uses $x$-values and $dy$ uses $y$-values.

Vocabulary

Solid of revolution

A three-dimensional solid formed by rotating a plane region around a line called an axis of rotation.

Disk method

A volume method that uses circular cross sections with radius $R$ and area $A = \pi R^2$.

Washer method

A volume method that uses ring-shaped cross sections with area $A = \pi(R^2 - r^2)$.

Shell method

A volume method that adds thin cylindrical shells using $dV = 2\pi p h\,dx$ or $dV = 2\pi p h\,dy$.

Radius function

A function that gives the distance from a slice to the axis of rotation.

Axis of rotation

The line around which a region is rotated to create a solid.