By This shell method calculator finds the volume of the solid you get by rotating the region under \( y = f(x) \), between \( x = a \) and \( x = b \), around the y-axis. It shows the integral \( \int_a^b x\,f(x)\,dx \) and the volume \( 2\pi \) times that, so you can compare each piece with your own work.
It is built for calculus students checking volume problems and for anyone who wants a y-axis volume without switching to \( x \) as a function of \( y \). Free, no sign-up.
How to use the shell method calculator
- Enter the height function \( f(x) \), the top of the region (the bottom is the x-axis). Use
^for powers,sqrt(),sin(),cos(),ln()ande^(x). Implicit multiplication like4xorx e^(-x)works. - Enter \( a \) and \( b \) with \( 0 \le a < b \);
piandpi/2are accepted. - Press Compute volume.
- Read \( \int_a^b x f(x)\,dx \) and the volume \( V \) in cubic units.
If you enter a negative \( a \), the calculator warns you: the formula below assumes the region sits to the right of the y-axis.
Formula
Cut the region into thin vertical strips. Spinning a strip at position \( x \) about the y-axis makes a thin cylindrical shell with radius \( x \), height \( f(x) \) and thickness \( dx \). Unrolled, the shell is nearly a flat sheet with volume \( 2\pi x \cdot f(x)\,dx \), so
$$V = 2\pi \int_a^b x\,f(x)\,dx$$
Read it as \( 2\pi \) times radius times height, integrated across the region. For a region between two curves, the height becomes \( f(x) - g(x) \); type that difference as the function (the area between curves calculator helps you find where the curves cross). The shell method guide covers rotation about other vertical lines, where the radius changes to something like \( 3 - x \).
Worked examples
Example 1 (a parabola). Rotate the region under \( y = 4x - x^2 \), \( 0 \le x \le 4 \), about the y-axis. The parabola touches the x-axis at 0 and 4, so those are the limits.
$$\begin{aligned}V &= 2\pi \int_0^4 x(4x - x^2)\,dx = 2\pi \int_0^4 (4x^2 - x^3)\,dx \\ &= 2\pi\left(\tfrac{256}{3} - 64\right) = 2\pi \cdot \tfrac{64}{3} = \tfrac{128\pi}{3}\end{aligned}$$
Enter 4x - x^2, 0 and 4 in the calculator above. It shows the integral as \( \frac{64}{3} \) and the volume as about 134.04129, which is \( \frac{128\pi}{3} \).
Example 2 (needs integration by parts). Rotate the region under \( y = \cos x \), \( 0 \le x \le \frac{\pi}{2} \), about the y-axis. The integral \( \int x\cos x\,dx \) calls for integration by parts: it equals \( x\sin x + \cos x \). So
$$V = 2\pi\left[x\sin x + \cos x\right]_0^{\pi/2} = 2\pi\left(\tfrac{\pi}{2} - 1\right) = \pi^2 - 2\pi$$
That is about 3.58642 cubic units. With the disk method you would have to solve \( y = \cos x \) for \( x \), which is much messier.
Shell or disk: which should you use?
Both give the same volume when set up correctly. Choose the one with the easier integral:
- Rotating about the y-axis with \( y = f(x) \) given: shells, because the strips are vertical and you keep everything in \( x \).
- Rotating about the x-axis: disks or washers usually win. Use the disk method calculator.
- The inverse function is hard or impossible to write: shells avoid it completely, as in Example 2.
A quick consistency check: rotate the triangle under \( y = 2 - x \), \( 0 \le x \le 2 \), about the y-axis. Shells give \( 2\pi\int_0^2 x(2 - x)\,dx = \frac{8\pi}{3} \). Disks in \( y \), with radius \( 2 - y \), give \( \pi \int_0^2 (2 - y)^2\,dy \), which is also \( \frac{8\pi}{3} \), a cone of radius 2 and height 2.
Related guides
- Shell Method for Volume: Formula, Examples and Disk vs Shell
- Disk and Washer Method: Volume of Revolution Step by Step
- Integration by Parts: Formula, LIATE Rule and Examples
Further reading
- Volumes of Solids of Revolution / Method of Cylinders (Paul’s Online Math Notes) — shells about vertical and horizontal axes, with four examples.
- Volumes of Solids of Revolution / Method of Rings (Paul’s Online Math Notes) — the disk and washer method, for comparison.
FAQ
What is the shell method formula?
\( V = 2\pi \int_a^b x\,f(x)\,dx \) for rotation about the y-axis, where \( x \) is the shell radius and \( f(x) \) its height.
Which axis does the shell method calculator rotate around?
The y-axis. For rotation about the x-axis, use the disk method, or write the curve as \( x \) in terms of \( y \) and apply shells with \( y \) as the variable.
Can I use it for a region between two curves?
Yes. Enter the top curve minus the bottom curve as \( f(x) \), for example x - x^3 on \( [0, 1] \), and the calculator uses that difference as the shell height.
Why does it warn me when a is negative?
The formula uses \( x \) as the radius, which only works when the region lies to the right of the y-axis. If part of it lies to the left, rotate each side separately or use symmetry.