Free Double Integral Calculator

What is a Double Integral?

A double integral extends the concept of single integration to two dimensions. It calculates the volume under a surface defined by a function z = f(x,y) over a region in the xy-plane. Double integrals are used in physics, engineering, and economics to model complex systems.

The basic form of a double integral is:

Where R is the region of integration, f(x,y) is the integrand function, and dA represents the area element.

How to Use This Calculator

1. Enter the function you want to integrate in the "Function" field (e.g., x² + y²)
2. Specify the limits of integration for x (a and b)
3. Enter the limits of integration for y as functions of x (u(x) and v(x))
4. Click "Calculate" to compute the double integral
5. View the result and visualization

Formula and Calculation

The double integral is calculated using the following steps:

1. Divide the region R into small subregions
2. Approximate the function over each subregion
3. Sum the areas of all subregions
4. Take the limit as the subregions become infinitesimally small

The calculator implements this numerically using the trapezoidal rule for integration.

Worked Example

Let's compute the double integral of f(x,y) = x² + y² over the region where 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.

The result is approximately 0.6667. The calculator will show this value along with a visualization of the function over the integration region.