Volume of Revolution: Step-by-Step Guide
When a region in the plane is rotated around an axis, the resulting 3D solid has a volume that can be computed by integration. The two main techniques are the disk/washer method (slice perpendicular to the axis of rotation) and the shell method (slice parallel to the axis). Choose whichever gives the simpler integral.
Math Racer skill guide · Calculus 2 · AP Calc BC / College
How to solve it, step by step
- Sketch the region and the axis of rotation. Draw the curves and shade the region. Identify whether you rotate about the x-axis, y-axis, or another horizontal/vertical line.\[\text{Region } R \text{ rotated about line } L\]
- Pick a method: disks/washers or shells. Use disks if slices perpendicular to the axis are solid disks; use washers if there is a hole; use shells if integrating along the axis-parallel direction is simpler.\[V_{disk} = \pi \int_a^b [R(x)]^2\,dx,\;\; V_{shell} = 2\pi\int_a^b r(x)\,h(x)\,dx\]
- Set up the integrand. For disks/washers, write outer radius R(x) and inner radius r(x). For shells, write radius (distance from axis) and height (top minus bottom). Determine the bounds.\[V = \pi\int_a^b \left([R(x)]^2 - [r(x)]^2\right)dx\]
- Integrate. Expand the integrand if needed and apply standard integration rules.\[V = \pi\bigl[F(b) - F(a)\bigr]\]
- Check units and reasonableness. Confirm the result is positive and roughly matches a bounding cylinder of radius R_max and height (b - a).\[0 < V \lesssim \pi R_{max}^2 (b-a)\]
Worked example
Find the volume of the solid generated by rotating y = sqrt(x) for x in [0, 4] about the x-axis.
Step 1: The region under y = sqrt(x) from x = 0 to x = 4 rotates about the x-axis. Slices perpendicular to the x-axis are solid disks of radius sqrt(x).
Step 2: Use the disk method since there is no hole.
Step 3: Set up V = pi * integral from 0 to 4 of (sqrt(x))^2 dx = pi * integral from 0 to 4 of x dx.
Step 4: Integrate: pi * [x^2/2] from 0 to 4 = pi * (16/2 - 0) = 8 pi.
Step 5: The volume is 8 pi cubic units, approximately 25.13.
Verification: A bounding cylinder of radius 2 (the maximum sqrt(4)) and height 4 has volume pi(2)^2(4) = 16 pi. Our solid is roughly half that, consistent with 8 pi.
Tips and common mistakes
- Always sketch the region and the axis before choosing a method.
- Disk method: V = pi * integral of [R(x)]^2 dx, where R(x) is the distance from axis to curve.
- Washer method: V = pi * integral of (R^2 - r^2) dx when the region does not touch the axis.
- Shell method: V = 2 pi * integral of r(x) h(x) dx, useful when the function is hard to invert.
- If rotating about y = k or x = k (not an axis), the radius becomes |y - k| or |x - k|.
- Always square radii correctly; (R - r)^2 is NOT R^2 - r^2.
- Match the variable of integration to slice direction: dx for vertical slices, dy for horizontal.
- Compare your answer to a rough cylinder volume as a sanity check.