Double Integral Calculator — Step-by-Step Solutions

The Double Integral Calculator evaluates integrals of functions of two variables over rectangular regions. It uses nested numerical integration to compute accurate results.

Enter your function of two variables (e.g. x and y), specify the bounds for each variable, and the calculator will evaluate the double integral. This is useful for finding volumes, surface areas, and other multi-dimensional quantities.

f(x, y)

x min

x max

y min

y max

Your calculations will appear here

How to Use the Double Integral Calculator

Enter a function of two variables (e.g. x^2 + y^2).
Specify the bounds for both variables (rectangular region).
The calculator uses nested Simpson's rule for accurate numerical evaluation.

Formula and Theory

A double integral extends the concept of integration to functions of two variables. It computes the volume under a surface z = f(x, y) over a region in the xy-plane.

For rectangular regions, the double integral can be evaluated as iterated integrals: first integrate with respect to one variable, then the other. The order of integration can be swapped (Fubini's theorem) for continuous functions.

Worked Examples

Example 1: Volume under a plane

Problem: Find the double integral of (x + y) over [0,1] x [0,1]

Solution: Inner: integral of (x+y)dy from 0 to 1 = x + 1/2. Outer: integral of (x + 1/2)dx from 0 to 1 = 1/2 + 1/2

Answer: 1

Frequently Asked Questions

What are double integrals used for?

Double integrals compute volumes, surface areas, mass of 2D regions with variable density, centres of mass, and moments of inertia.

Does the order of integration matter?

For continuous functions over rectangular regions, the order does not affect the result (Fubini's theorem). For more complex regions, the bounds may change when swapping order.