Estimate rotational volume from sampled radius functions and bounds. Review disks, areas, and integration instantly. Export tables, graphs, and reports for deeper analytical study.
The disk method treats the solid as many thin circular slices. Each slice has area A(t) = π[R(t)]², where R(t) is the radius at position t.
The total volume is V = π∫ab[R(t)]²dt. This file evaluates the integral numerically with Simpson, trapezoidal, or midpoint integration.
For physics use, optional mass is m = ρV. Optional axial moment of inertia is I = 1/2 ρπ∫ab[R(t)]4dt. Optional rotational kinetic energy is K = 1/2 Iω².
| Example Radius Model | Bounds | Method | Intervals | Density | Angular Speed | Volume |
|---|---|---|---|---|---|---|
| R(x) = 2 + 1x | 0 to 3 m | Simpson | 12 | 1.2 | 3 rad/s | 122.522113 m3 |
This calculator estimates the volume of solids formed by rotating a radius profile around an axis. The polynomial model makes it useful for lab approximations, prototype shaping, motion studies, and educational demonstrations where exact antiderivatives are not always convenient.
Simpson integration usually gives the strongest accuracy for smooth curves. Trapezoidal and midpoint methods are included because they help compare numerical behavior and verify sensitivity to interval count. That makes the page useful for both computation and teaching.
The optional physics outputs extend the calculator beyond geometry. When density is known, the tool returns mass. When angular speed is entered, it also estimates axial inertia and rotational kinetic energy. These values help connect geometric solids with practical mechanics.
The graph reveals how radius changes across the interval. The slice table shows representative disks so users can inspect local areas rather than trusting a single final number. Exports make it easier to save worked results for reports, assignments, or technical notes.
It calculates the volume of a solid created by revolving a radius function around an axis. The solid is modeled as many thin circular disks.
They let you compare accuracy and stability. Simpson is often best for smooth curves, while trapezoidal and midpoint are useful for checks and simpler approximations.
It uses a fourth degree polynomial radius model. You enter coefficients c0 through c4, and the calculator evaluates the radius across the chosen interval.
Simpson integration requires an even number of intervals. If you enter an odd value, the calculator adds one interval automatically and shows a note.
Yes. Add density to estimate mass. Add angular speed too, and the tool also estimates axial moment of inertia and rotational kinetic energy.
The calculator uses the absolute radius magnitude because physical disk radius cannot be negative. Volume and area are therefore based on nonnegative slice radii.
It plots the sampled radius and corresponding disk area over the interval. This helps you see where the solid is narrow, wide, or changing rapidly.
They include the summary metrics and representative slice data. This makes the output easier to archive, share, review, or attach to coursework.
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.