Compute volume from radius functions with numerical integration. Export slice data and verify setup quickly. Designed for students, engineers, and careful problem checking today.
Example setup: y = x + 1, a = 0, b = 2, and 4 midpoint slices.
| Slice | x midpoint | Radius | Disk area | Slice volume |
|---|---|---|---|---|
| 1 | 0.25 | 1.25 | 4.908739 | 2.454369 |
| 2 | 0.75 | 1.75 | 9.621128 | 4.810564 |
| 3 | 1.25 | 2.25 | 15.904313 | 7.952156 |
| 4 | 1.75 | 2.75 | 23.758294 | 11.879147 |
The disk method around the x-axis uses V = π ∫[a to b] (r(x))² dx.
Here, r(x) is the radius from the axis to the curve.
Each slice has area πr² and thickness Δx.
The calculator estimates total volume with midpoint, trapezoidal, or Simpson rules.
The disk method finds the volume of a solid of revolution. A curve rotates around the x-axis. Each thin slice becomes a disk. The calculator adds many disk volumes. That gives a strong numerical estimate. This approach is useful in physics and engineering. It helps with tanks, lenses, nozzles, and symmetric parts.
Every slice has a tiny thickness. Its radius comes from the function value. The cross-sectional area is πr². Multiply area by thickness to get a tiny volume. Add all tiny volumes across the interval. Calculus writes this as an integral. Numerical integration lets the page solve it fast.
This tool supports several common radius models. You can test a polynomial curve. You can also use a power model. Exponential growth is available too. Sine and cosine options are included. These choices cover many classroom exercises. They also fit many practical profiles. The slice table shows how radius changes with x.
The result section gives the estimated volume first. It also shows interval width and slice count. A sample slice table appears below. That table helps you inspect the setup. You can catch sign mistakes early. You can also review area values. Export buttons let you save the result. This is helpful for reports and assignments.
Many physical objects are modeled by rotation. Pressure vessels often use curved ends. Optical parts may follow smooth profiles. Pipes and ducts can widen gradually. Disk estimates help compare design choices. They also support quick checks before detailed simulation. Students can connect geometry with measurable volume. That makes the integral more meaningful.
Choose bounds that match the real shape. Use more slices for better accuracy. Check whether the radius function stays realistic. Large coefficients may create huge results. Trigonometric models need careful units. This calculator assumes radians in the trig functions. Review the table before exporting. Small checks prevent large interpretation errors later.
Because the calculator is interactive, you can compare several models in minutes. Try changing only one parameter at a time. That habit shows which input drives volume the most. It also improves intuition during study and design review sessions.
It estimates the volume of a solid formed by rotating a curve around the x-axis. Each thin slice is treated as a circular disk.
Different methods approximate the integral in different ways. Midpoint is simple and reliable. Simpson often gives higher accuracy when the function is smooth.
Pick a function family and enter its coefficients. The calculator uses the function value as the radius magnitude for each sampled x position.
Yes. The tool uses the absolute value as the disk radius. Squaring the radius keeps the cross-sectional area positive.
Simpson works by grouping slices into paired intervals. An odd count breaks that pattern, so the method needs an even number of slices.
No. The trigonometric inputs are treated in radians. Convert degree-based expressions before using those function options.
The result is shown in cubic form of your unit label. If you enter cm, the result appears as cm³.
Increase slices when the curve changes quickly or when you want a tighter estimate. More slices usually improve numerical accuracy.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.