Estimate the volume of a solid formed by rotating y = f(x) around the x-axis. This page supports Simpson’s rule, trapezoidal rule, and midpoint rule for precise numerical integration.
Function input examples: sqrt(x), x^2, sin(x), 2*x+1, exp(-x). Use * for multiplication.
Calculator Form
Example Data Table
These example cases show common solids of revolution that work well with the disk method.
| Example | Function | Interval | Rotation | Exact setup | Reference value |
|---|---|---|---|---|---|
| Parabola root profile | y = sqrt(x) | [0, 4] | x-axis | V = π∫₀⁴ x dx | 8π ≈ 25.132741 |
| Linear cone profile | y = x | [0, 2] | x-axis | V = π∫₀² x² dx | 8π/3 ≈ 8.377580 |
| Descending line | y = 2 - x | [0, 2] | x-axis | V = π∫₀² (2-x)² dx | 8π/3 ≈ 8.377580 |
| Trig example | y = sin(x) | [0, π] | x-axis | V = π∫₀π sin²(x) dx | π²/2 ≈ 4.934802 |
Formula Used
The disk method computes volume by slicing the solid into very thin circular disks perpendicular to the x-axis. Each disk has area A(x) = π[r(x)]², where the radius is the distance from the curve to the axis.
For this calculator, the solid is generated by rotating y = f(x) around the x-axis. The numerical engine evaluates:
Here g(x) = π[f(x)]² and h = (b - a) / n.
How to Use This Calculator
- Enter the function as y = f(x).
- Set the lower bound a and upper bound b.
- Choose the number of intervals for the numerical approximation.
- Select Simpson, trapezoidal, or midpoint integration.
- Optionally change the plot density and displayed unit label.
- Press Calculate Volume by Disks.
- Review the result block above the form, graph, and slice table.
- Download a CSV or PDF report for sharing or checking later.
Frequently Asked Questions
1) What does this calculator measure?
It estimates the volume of a solid formed by rotating a curve around the x-axis. The tool squares the radius, multiplies by π, and integrates across the chosen interval.
2) When should I use the disk method?
Use the disk method when the solid has no hole along the axis of rotation. If there is an inner radius, the washer method is usually more appropriate.
3) Which numerical method is best?
Simpson’s rule is usually the most accurate for smooth functions. Trapezoidal is simple and stable. Midpoint is fast and often better than trapezoidal for curved profiles.
4) Why does the result stay positive for negative curve values?
Radius represents distance from the axis, so it cannot be negative in geometry. The formula uses the square of the radius, which makes the disk area nonnegative.
5) What function syntax is supported?
You can use x, numbers, parentheses, +, -, *, /, ^, and functions such as sin, cos, tan, sqrt, abs, exp, log, ln, pow, floor, and ceil.
6) Why did the interval count change?
Simpson’s rule requires an even number of subintervals. If you enter an odd count, the calculator automatically increases it by one and explains that adjustment.
7) How can I improve accuracy?
Increase the number of intervals, keep the function continuous on the chosen interval, and prefer Simpson’s rule when the curve is smooth and well-behaved.
8) What do the CSV and PDF files include?
The exports include the function, bounds, method, main result values, and the displayed numerical slice table so you can review or share the calculation clearly.