AP® Calculus AB · Unit 8 · Topic 8.9
Volume with the Disc Method: Revolving Around the x- or y-Axis
The disc method finds the volume of a solid formed by spinning a curve around an axis by stacking up infinitely many thin circular discs and adding their volumes with an integral.
Picture the region under a curve y = f(x), between x = a and x = b. Spin that region around the x-axis and it sweeps out a solid shape, like a vase or a football. Slice that solid into thin vertical pieces and each piece looks like a coin: a short cylinder, or disc. Each disc has some tiny thickness dx and a circular face. If you know the area of that circular face at every x, you can add up all the discs with an integral to get the total volume.
The key move is finding the radius of each disc. When you revolve around the x-axis, the radius at any point x is just the height of the function there, f(x), because that is the distance from the axis of rotation out to the curve. The area of that circular face is π times radius squared, so π times f(x) squared. Stack these areas along the axis with an integral and you get volume equals π times the integral from a to b of f(x) squared dx. Revolving around the y-axis works the same way, except now you need x as a function of y, and you integrate with respect to y instead.
This is the simplest case of the broader slicing idea that runs through this unit: every volume problem is really an integral of cross-sectional area. The disc method assumes the region touches the axis of rotation directly, so the radius is a single distance from axis to curve. When the spinning region is lifted off the axis, leaving a gap, each slice becomes a washer, a disc with a hole punched out, and you subtract the inner area from the outer one. That is the natural next step after mastering discs.
Key terms
- Disc method
- A technique that finds volume by integrating the area of circular cross-sections perpendicular to the axis of rotation.
- Radius of revolution
- The distance from the axis of rotation to the curve, which becomes the radius of each disc.
- Cross-sectional area
- The area of the flat slice you get by cutting the solid perpendicular to the axis at a given point.
- Washer method
- A version of the disc method used when the region is lifted off the axis, so each slice has a hole and you subtract the inner area.
Watch it explained
- How it worksHow does each disc's radius come straight from the height of the function?
- Worked exampleFind the volume when y = √x from 0 to 4 is revolved around the x-axis.
- Common mistakeWhy isn't the disc radius the distance between two curves in this setup?
- Ties togetherHow does punching a hole in a disc turn the disc method into the washer method?
Check yourself
End of topic.