Calculator Inputs
Enter boundary functions, bounds, and axis location. Supported functions include sin, cos, tan, sqrt, abs, exp, log, log10, and ln.
Formula Used
General shell method formula
V = 2π ∫ radius(u) × height(u) du
The shell variable u is x for vertical shells and y for horizontal shells.
Vertical shells about x = c
V = 2π ∫[a,b] |x - c| · |f(x) - g(x)| dx
Use this when your region is described by top and bottom functions in terms of x.
Horizontal shells about y = c
V = 2π ∫[a,b] |y - c| · |f(y) - g(y)| dy
Use this when your region is described by right and left functions in terms of y.
Numerical method
This calculator evaluates shell contributions across evenly spaced points and estimates volume with Simpson integration. A trapezoidal estimate is also shown for comparison.
How to Use This Calculator
- Choose the shell orientation that matches your axis of rotation.
- Enter the two boundary expressions using x or y only.
- Set the lower and upper bounds for the shell variable.
- Enter the axis offset c for x = c or y = c.
- Choose an even number of intervals for better accuracy.
- Press Calculate Volume to view the result above the form.
- Review the graph, shell table, and cross-check metrics.
- Download CSV or PDF if you need a saved report.
Example Data Table
Example region: rotate y = x² and y = 0 on 0 ≤ x ≤ 2 about the y-axis. Here, radius = x and height = x², so the exact volume is 8π ≈ 25.132741.
| x | y = x² | y = 0 | Radius x | Height x² | 2πxh |
|---|---|---|---|---|---|
| 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
| 0.5000 | 0.2500 | 0.0000 | 0.5000 | 0.2500 | 0.7854 |
| 1.0000 | 1.0000 | 0.0000 | 1.0000 | 1.0000 | 6.2832 |
| 1.5000 | 2.2500 | 0.0000 | 1.5000 | 2.2500 | 21.2058 |
| 2.0000 | 4.0000 | 0.0000 | 2.0000 | 4.0000 | 50.2655 |
FAQs
1. What does the shell method measure?
It measures the volume of a solid formed by rotating a planar region around an axis. The method adds cylindrical shell contributions across the chosen interval.
2. When should I use shells instead of washers?
Use shells when the natural boundary functions align with shells directly. This often avoids solving for inverse functions and keeps the integral simpler.
3. Why are absolute values used for radius and height?
Radius and height represent physical lengths, so they should not be negative. Absolute values keep the geometry valid even if expressions are entered in reversed order.
4. What functions can I enter?
You can enter algebraic and common transcendental expressions using operators and functions like sin, cos, tan, sqrt, abs, exp, log, log10, and ln.
5. Why does the calculator show two volume estimates?
Simpson integration is the main estimate. The trapezoidal result gives a quick consistency check. A small percentage difference usually suggests a stable numerical approximation.
6. How many intervals should I use?
Smooth functions often work well with 100 to 400 intervals. Increase intervals when the graph changes rapidly, has steep slopes, or you want stronger numerical agreement.
7. Can I rotate around shifted axes like x = 3 or y = -2?
Yes. Enter the axis offset in the axis value field. The calculator then computes shell radius from the distance to that shifted line.
8. What does the graph show?
The graph shows the two boundary functions and the shell integrand 2πrh. This helps you see where shell contributions are small, large, or changing quickly.