Recipe

Choose and set up a volume of revolution

Use when: a flat region is revolved around an axis and the volume of the resulting solid is needed

Goal: the correctly chosen method (disk, washer, or shell), with its integral fully set up in the variable that avoids unnecessary inversion

Taught in Choosing a volume method

1. 1

   Sketch the flat region and the axis of revolution together. Check whether the region touches the axis along one whole edge, or is lifted away from it by a second boundary curve.

   Example

   The region under y=2x down to y=x^2 on [0,2], revolved about the y-axis, is lifted away from the y-axis itself (both boundary curves sit at positive x), which rules out a plain disk.

   • the region touches the axis along one edge — each cross-section is a full disk; no hole.
   • the region is lifted away from the axis by a second boundary — each cross-section is a washer, an outer disk with a hole punched through it.

   Why: this single check decides whether an extra inner-radius term is needed at all before any formula is written down.

2. 2

   See whether every boundary curve is already a clean function of the variable matching the axis of revolution (for disks/washers), or of the variable perpendicular to it (for shells).

   Example

   Both y=2x and y=x^2 are already clean functions of x, but the axis of revolution here is the y-axis (vertical), so disks/washers around it would need both curves inverted to functions of y. Shells avoid that inversion entirely.

   • every boundary is already a function of x, and the axis is vertical — shells, integrating in dx but built from vertical strips, skip inverting to functions of y.
   • every boundary is already a function of x, and the axis is horizontal — disks or washers, integrating in dx, are the natural choice with no inversion needed.

   Why: picking the method that matches the boundary's natural variable is what turns a short setup into a long one or vice versa; the final volume is identical either way.

3. 3

   For disks/washers, the radius is the distance from the axis to the outer curve (and, for a washer, the inner curve too). For shells, identify the strip's height and its radius (distance from the axis to the strip).

   Example

   For shells about the y-axis: at position x, the strip runs from g(x)=x^2 up to f(x)=2x, so its height is f(x)-g(x)=2x-x^2 and its radius is x itself.

   Why: mixing up radius and height, or outer and inner radius, produces the single most common setup error in this topic.

4. 4

   Plug the radius (and height, or inner/outer radii) into the disk, washer, or shell formula, matched to the axis.

   Example

   V = 2\pi\int_0^2 x(2x-x^2)\,dx

   Why: this is just the slicing idea from earlier in the chapter, specialized to whichever cross-section shape Step 1 identified.

5. 5

   Antidifferentiate, evaluate at the bounds, and confirm the result is a positive volume.

   Example

   2\pi\int_0^2(2x^2-x^3)\,dx = 2\pi\left[\frac{2x^3}{3}-\frac{x^4}{4}\right]_0^2 = 2\pi\left(\frac{16}{3}-4\right) = 2\pi\cdot\frac43 = \frac{8\pi}{3}

   ⚡ Shortcut

   Before committing to a method, ask only "does inverting a boundary curve to solve for the other variable look easy or ugly?" Shells never require that inversion; disks/washers around a vertical axis always do. That one question usually settles the choice faster than sketching every cross-section shape.

   Why: a negative or zero result is a sign that the height or radius was set up backwards, the same check every volume method needs.