Volume by Disks
Volume of a solid of revolution about the x-axis.
Formula
Variables
Example
Rotate a curve to make a solid
How it works
This formula adds up the volumes of very thin circular slices of a solid formed by rotating a graph around the x-axis. At each x-value, the radius of the slice is f(x), so its cross-sectional area is \u03c0[f(x)]^2; integrating those areas from a to b accumulates the total volume. It works because a shape that is curved and continuous can be approximated by many tiny disks, and the integral is the exact limit of that summation.
When to use it
Use this when a region under a curve is revolved around the x-axis and you want the volume of the resulting solid. It is especially useful in physics, engineering, and geometry problems involving axially symmetric objects like domes, tanks, funnels, or machined parts.
Worked example, step by step
Find the volume when the region under f(x) = 2 - x on 0 \u2264 x \u2264 2 is rotated about the x-axis. V = \u03c0\int_0^2 [f(x)]^2 dx V = \u03c0\int_0^2 (2 - x)^2 dx V = \u03c0\int_0^2 (4 - 4x + x^2) dx V = \u03c0\left[4x - 2x^2 + \frac{x^3}{3}\right]_0^2 V = \u03c0\left((8 - 8 + \frac{8}{3}) - 0\right) V = \frac{8\pi}{3} So the volume is \frac{8\pi}{3} cubic units.
Common mistakes to avoid
- Forgetting to square the radius function before integrating
- Using the wrong interval or swapping the bounds without fixing the sign
- Applying the disk method when the solid has a hole and a washer method is needed
Did You Know?
The disk method builds a solid from infinitely thin circular slices, exactly how a lathe shapes wood.
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