Calculate the volume of a solid of revolution using the disk or washer method with Simpson's rule numerical integration. Supports linear, quadratic, square root, sine, and custom polynomial functions.
A solid of revolution is a three-dimensional shape formed by rotating a two-dimensional curve around an axis. This fundamental concept from integral calculus allows you to compute volumes of shapes like spheres, cones, paraboloids, toroids, and any surface generated by revolving a function f(x) around a line.
The disk method applies when the solid has no hole: you slice it into thin circular disks of radius R(x) = |f(x) − k| (where y = k is the axis of rotation), each with area πR², and integrate from a to b: V = π ∫ₐᵇ R(x)² dx. The washer method extends this to shapes with a hole: you have an outer function f(x) and an inner function g(x), and V = π ∫ₐᵇ [R(x)² − r(x)²] dx, where R is the outer radius and r is the inner radius.
This calculator uses Simpson's rule — a highly accurate numerical integration method — to evaluate the integral. Simpson's rule approximates the integrand using parabolic arcs over each pair of subintervals, achieving fourth-order accuracy (error ~ O(h⁴)). You can control the number of subdivisions for higher precision.
Five built-in function types are supported: linear (mx + b), quadratic (ax² + bx + c), square root (a√x), sine (a·sin(bx)), and custom polynomial (c₀ + c₁x + c₂x² + c₃x³). The washer method allows you to specify both outer and inner functions independently. A cross-section table shows the disk/washer area at each sample point, and a radius profile visualization shows how the solid's radius changes across the interval.
Classic presets include generating a sphere from a semicircle, a cone from a linear function, a paraboloid, and a washer-method example with two different functions.
Volume-of-revolution problems combine function setup, radius interpretation, and definite integration, so most mistakes happen before the integral is even evaluated. This calculator helps you test the geometry of a disk or washer setup, compare built-in function families, and see how the chosen interval and axis of rotation affect the final solid, which is especially useful in calculus classes and engineering modeling.
Disk method: V = π ∫ₐᵇ [f(x) − k]² dx Washer method: V = π ∫ₐᵇ ([f(x) − k]² − [g(x) − k]²) dx Simpson's rule: ∫ₐᵇ h(x) dx ≈ (Δx/3)[h(x₀) + 4h(x₁) + 2h(x₂) + 4h(x₃) + ... + h(xₙ)] where Δx = (b − a)/n and n is even
Result: Volume ≈ 20.1062 (= 32π/5)
Revolving y = x² from x = 0 to x = 2 around the x-axis: V = π ∫₀² (x²)² dx = π ∫₀² x⁴ dx = π[x⁵/5]₀² = π(32/5) = 32π/5 ≈ 20.1062. Simpson's rule with 100 subdivisions gives this result to high accuracy.
The real decision in a volume-of-revolution problem is whether each cross-section is a full disk or a washer with a hole. If the region touches the axis of rotation, the inner radius is zero and the disk method is enough. If the region stays away from the axis or is bounded by two curves, you need a washer setup so the inner radius is subtracted from the outer radius before integrating.
Most setup errors happen when students confuse the function value with the radius. The radius is the distance from the curve to the axis of rotation, so revolving around $y = k$ means the effective radius is $|f(x) - k|$. This calculator makes that explicit in the cross-section table, where you can inspect $R$, $r$, and the resulting area at sample points instead of treating the integral as a black box.
Even when an antiderivative exists, Simpson's rule is valuable because it lets you test a setup quickly and compare exact reasoning against a numerical result. That is especially helpful for sinusoidal or custom polynomial profiles, where the geometry matters as much as the algebra. The radius profile also shows how the solid changes across the interval, which is useful for understanding why certain regions produce spheres, cones, paraboloids, or more unusual bodies.
The disk method computes the volume of a solid of revolution by slicing it into thin circular disks perpendicular to the axis of rotation. Each disk has area πR² and thickness dx. The volume is V = π ∫ R² dx.
The washer method extends the disk method to solids with a hole (like a donut cross-section). Each slice is a washer with outer radius R and inner radius r: V = π ∫ (R² − r²) dx.
Simpson's rule is a numerical integration method that approximates the integrand with parabolic arcs. It uses the formula: ∫ ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + ... + f(xₙ)]. It is highly accurate — exact for polynomials up to degree 3.
Revolve f(x) = √(r² − x²) around the x-axis from −r to r. In custom mode, set c₀ = r², c₂ = −1 (note: the function evaluates √(c₀ + c₂x²) effectively via integrating the squared value). Use the sphere preset.
Yes — select "y = k" as the axis of rotation and enter the value of k. The radius becomes |f(x) − k| instead of |f(x)|.
The calculator takes the absolute value of the integral, so volume is always non-negative. If you get zero, check that f(x) actually produces nonzero values on the interval [a, b] and that b > a.