Double Integral Calculator

A double integral ∫∫_R f(x,y) dA computes the volume under the surface z = f(x,y) above a region R in the xy-plane. For a rectangular region [a,b] × [c,d], it can be evaluated as an iterated integral:

∫_a^b ∫_c^d f(x,y) dy dx

Inner integral first (treating x as constant), then outer integral.

This calculator uses Simpson’s rule in two dimensions. For each of n+1 evenly spaced x values, it integrates f(x,y) over y using 1D Simpson’s rule, giving a function g(x). Then it integrates g(x) over the x range, again with Simpson’s rule. The combined error is O(h^4) where h is the step size — accurate to about 6 decimal places for smooth integrands with n = 40.

Three integrand forms are available:

Sum: f(x,y) = ax^p + by^q. Easily verifiable by hand for small p, q. Product: f(x,y) = ax^p * y^q. Factors into ∫x^p dx · ∫y^q dy when integrated over a rectangle. Constant: f(x,y) = a. Integral equals a × (area of rectangle).

Worked example: ∫_0^2 ∫_0^1 (x² + y²) dy dx. Inner: ∫_0^1 (x² + y²) dy = x²·y + y³/3 from 0 to 1 = x² + 1/3. Outer: ∫_0^2 (x² + 1/3) dx = x³/3 + x/3 from 0 to 2 = 8/3 + 2/3 = 10/3 ≈ 3.333.

Use cases: volume under a surface, mass with variable density, average value of a function over a region, center of mass coordinates, moments of inertia. In probability, double integrals compute probabilities for joint continuous distributions.

For non-rectangular regions you would substitute variable bounds (Type I or Type II regions) or change to polar/elliptic coordinates with a Jacobian — beyond the scope of this fixed-rectangle calculator.