Volume of a Solid of Revolution
Enter your functions and interval to estimate volume using Simpson’s Rule.
Shell: V = 2π∫ab(radius)(height)dx
Supported functions: sin, cos, tan, sqrt, abs, log, ln, exp, constants pi and e.
What Is a Solid of Revolution?
A solid of revolution is a 3D shape formed by rotating a 2D curve or region around an axis. If you spin a line around the x-axis or y-axis, the sweep of that motion creates volume. Classic examples include spheres, cones, and torus-like forms.
In calculus, we usually compute this volume using one of two frameworks: the disk/washer method or the shell method. Both are equivalent when set up correctly, but one often leads to cleaner algebra than the other depending on the function and axis.
How This Revolution Solid Calculator Works
1) Disk / Washer Method
Use this when cross-sections perpendicular to the axis of rotation are natural.
You provide an outer radius R(x) and inner radius r(x), then the tool evaluates:
V = π ∫(R(x)^2 - r(x)^2) dx
- If
r(x) = 0, you have disks. - If
r(x) > 0, you have washers (a hole in the center).
2) Cylindrical Shell Method
Use this when slices parallel to the axis are easier to describe. You input a radius function and a height function, and the calculator evaluates:
V = 2π ∫(radius × height) dx
This is especially useful for rotation around the y-axis when your function is already written as y = f(x).
Input Tips for Accurate Results
- Use
^for exponents (example:x^3). - Parenthesize clearly (example:
(x+1)^2). - Set reasonable bounds where the geometry is defined.
- Increase subdivisions for better accuracy on curved or oscillating functions.
Example Problems You Can Try
Example A: Disk Method
Rotate y = x^2 + 1 about the x-axis on [0, 2].
Set R(x)=x^2+1, r(x)=0, a=0, b=2.
Example B: Washer Method
Rotate region between y=3 and y=1 on [0,4].
Set R(x)=3, r(x)=1, and bounds 0 to 4.
Example C: Shell Method
Rotate region under y=x^2+1 around the y-axis from 0 to 2.
Set shell radius to x, shell height to x^2+1.
Common Mistakes
- Mixing up inner and outer radius in washer setups.
- Using negative shell heights from incorrect region definitions.
- Forgetting unit consistency (length units produce cubic units for volume).
- Using too few subdivisions on complicated functions.
Final Notes
This calculator gives a strong numerical estimate and is ideal for checking homework, building intuition, and validating symbolic integration. For piecewise regions or discontinuities, split the interval and calculate each part separately for best reliability.