Volumes of Revolution - Disk and Washer Method

Rotating Regions Around an Axis

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The disk and washer methods are calculus tools for finding the volume of a three dimensional solid formed by rotating a two dimensional region around a line. They matter because many real objects, from machine parts to containers, can be modeled as solids of revolution. Instead of adding up tiny rectangular areas, we add up many thin circular slices. This turns a geometric picture into an integral that gives exact volume.

The disk method is used when each slice is a solid circle, so there is no hole in the middle. The washer method is used when each slice has an outer radius and an inner radius, creating a ring shaped cross section. In both cases, the key step is identifying the radius or radii as distances from the curve to the axis of rotation. Once the radii are written as functions, the volume comes from integrating cross sectional area over the interval.

Key Facts

Disk method: $V = \pi \int_a^b [R(x)]^2 \,dx$
Washer method: $V = \pi \int_a^b ([R(x)]^2 - [r(x)]^2) \,dx$
If rotating $y = f(x)$ about the x-axis, then $R(x) = f(x)$
If rotating $y = f(x)$ about $y = k$ , then radius = $|f(x) - k|$
If integrating with respect to $y$ , then $V = \pi \int_c^d ([R(y)]^2 - [r(y)]^2) \,dy$
Cross sectional area for a washer: $A = \pi(R^2 - r^2)$ , and for a disk: $A = \pi R^2$

Vocabulary

Solid of revolution: A three dimensional shape formed by rotating a two dimensional region around a line.
Axis of rotation: The line about which a region is rotated to create a solid.
Radius: The distance from the axis of rotation to the outer edge of a disk or washer slice.
Inner radius: The distance from the axis of rotation to the inner edge of a washer, which creates the hole.
Cross section: A thin slice of the solid perpendicular to the direction of integration, used to build the volume integral.

Common Mistakes to Avoid

Using the disk formula when there is a hole, which is wrong because a washer needs both an outer radius and an inner radius so the missing middle is subtracted.
Forgetting that radius is a distance to the axis of rotation, which is wrong because the radius is not always just the function value and may require subtracting the axis location.
Integrating with respect to x when the slices are naturally horizontal, which is wrong because the variable of integration must match how the cross sections are taken.
Not squaring both radii in the washer formula, which is wrong because area depends on $\pi$ times radius squared, so subtracting unsquared radii gives the wrong value.

Practice Questions

1 Find the volume formed by rotating the region under $y = x$ from $x = 0$ to $x = 3$ about the $x$ -axis using the disk method.
2 Find the volume formed by rotating the region between $y = 4$ and $y = x^2$ from $x = 0$ to $x = 2$ about the $x$ -axis using the washer method.
3 A region is rotated about the line $y = 2$ and the graph lies partly above and partly below that line. Explain how you would determine the correct radius or radii for a disk or washer integral.