Volume of Revolution Calculator
Calculate volumes of solids formed by rotating curves around axes using disk and shell methods
Examples: sqrt(x) from 0 to 4, x^2 from 0 to 2
Results
Enter values and click Calculate to see the result.
Theory & Formula
Volume of Revolution
When a curve is rotated around an axis, it forms a three-dimensional solid. The volume can be calculated using the disk method or shell method.
Disk Method
Used when rotating a region bounded by f(x) and the x-axis. Each cross-section is a disk.
\(V = \pi \int_a^b [f(x)]^2 \, dx\)Washer Method
Used when rotating a region between two functions. Each cross-section is a washer (disk with a hole).
\(V = \pi \int_a^b ([R(x)]^2 - [r(x)]^2) \, dx\)Shell Method
Used when rotating about a vertical axis. Each shell has radius x and height f(x).
\(V = 2\pi \int_a^b x \cdot f(x) \, dx\)Example
Find the volume when y = √x from x=0 to x=4 is rotated about the x-axis:
\(f(x) = \sqrt{x}, \quad [0, 4]\)\(V = \pi \int_0^4 (\sqrt{x})^2 \, dx = \pi \int_0^4 x \, dx\)\(V = \pi \left[\frac{x^2}{2}\right]_0^4 = \pi \cdot 8 = 8\pi \approx 25.13\)