🏺 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.

drag to orbit · scroll / pinch to zoom · double-click resets

Outer curve f(x) =

Inner curve g(x) = (optional, washers)

From x = a

To x = b

Axis of rotation

Sweep 360°

Density (optional, g/cm³ with units = cm)

Slices shown 12

Volume—

Volume ÷ π—

Method—

Outer surface area—

Centroid along axis—

Area of region (Pappus check)—

Mass · capacity—

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) dx with 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.