Double Integral Calculator

Easily calculate the double integral of a function with step-by-step solutions. Just enter your function and the limits of integration.

Function f(x, y):

Enter the function in terms of x and y. Use ^ for power, e.g., x^2, sin(x), cos(y), etc.

Inner Lower Bound (y):

Inner Upper Bound (y):

Outer Lower Bound (x):

Outer Upper Bound (x):

Result:

Calculation Steps:

What is a Double Integral?

A double integral is used to calculate the volume under a surface in three-dimensional space, just as a single integral calculates the area under a curve in two-dimensional space. Imagine you have a surface defined by a function f(x, y) over a region in the xy-plane. The double integral of f(x, y) over this region gives you the volume between the surface and the xy-plane.

In simpler terms, if you think of f(x, y) as the height at each point (x, y), then the double integral sums up all these heights over a given area to find the total "volume". Double integrals are fundamental in multivariable calculus and have applications in physics, engineering, and statistics for problems involving areas, volumes, and average values in two dimensions.

For example, to calculate the volume under the surface z = x² + y² over the square region [0, 1] x [0, 1] in the xy-plane, you would use a double integral. This calculator helps you solve these integrals numerically, providing an approximate solution.