🏺 Solids of Revolution Volume Calculator 3D
🏺 Solids of Revolution Volume Calculator 3D
Spin any curve y = f(x) around the x- or y-axis and watch the 3D solid form. Get the volume by disc/washer and shell methods, the surface area and centroid, with an optional inner curve for washers. Slices show exactly what the integral adds up.
How this works — discs, washers and shells
Rotating the region between g(x) ≤ y ≤ f(x), a ≤ x ≤ b sweeps out a solid. Slice it and add up the slices:
- Washers (x-axis): each slice is a ring of outer radius f and inner radius g, so
V = π ∫ₐᵇ (f² − g²) dx. With no inner curve it is the disc method. - Shells (y-axis): each thin strip at distance x becomes a cylindrical shell of height f − g, so
V = 2π ∫ₐᵇ x (f − g) dx(with |x| used if the region crosses the axis). - Surface area of the outer skin:
2π ∫ r(x) √(1 + f′(x)²) dxwith r = |f| (x-axis) or |x| (y-axis). End caps are not included. - Pappus's theorem is shown as a cross-check: volume = region area × distance travelled by its centroid (2π r̄).
Integrals use composite Simpson's rule with 2,000 panels (and a central-difference derivative for arc length), which is accurate to many digits for smooth curves. Worked example: y = √x from 0 to 4 about the x-axis gives V = π ∫₀⁴ x dx = 8π ≈ 25.1327.
Functions: sin cos tan exp ln log10 sqrt abs and more, ^ for powers, constants pi and e. Parsing is done by a built-in parser — nothing is ever run with eval.