Double Integral Calculator Symbolab

A double integral is a mathematical concept used to calculate the volume under a surface or the area of a region in two-dimensional space. This calculator helps you compute double integrals using Symbolab's powerful computational engine.

What is a Double Integral?

A double integral extends the concept of a single integral to two dimensions. It calculates the volume under a surface defined by a function z = f(x,y) over a region D in the xy-plane. Double integrals are fundamental in physics, engineering, and economics for calculating quantities like mass, probability, and work.

The double integral of a function f(x,y) over a region D is written as:

Double Integral Notation

∫∫_D f(x,y) dA

This represents the integral of f(x,y) over the area D in the xy-plane.

Double Integral Formula

The double integral can be computed using iterated integrals when the region D can be described as a Cartesian product of intervals:

Iterated Integral Formula

∫_a^b ∫_c(x)^d(x) f(x,y) dy dx

Where:

f(x,y) is the integrand function
D is the region of integration
a and b are the limits of integration for x
c(x) and d(x) are the limits of integration for y as functions of x

How to Calculate a Double Integral

Calculating a double integral involves these steps:

Identify the region D of integration
Determine the limits of integration for x and y
Set up the iterated integral
Integrate with respect to y first, then x
Evaluate the definite integrals

Example

Calculate ∫∫_D (x² + y²) dA where D is the rectangle [0,2] × [0,3].

Solution: ∫₀² ∫₀³ (x² + y²) dy dx = 24

Applications of Double Integrals

Double integrals have numerous practical applications:

Calculating mass and center of mass of a lamina
Finding probabilities in continuous probability distributions
Computing work done by a variable force
Determining the area of a region in polar coordinates
Modeling heat distribution in physics

FAQ

What is the difference between single and double integrals?

A single integral calculates the area under a curve in one dimension, while a double integral calculates the volume under a surface in two dimensions.

When would I use a double integral instead of a single integral?

Use double integrals when dealing with quantities that vary over a two-dimensional region, such as mass distribution, probability density, or heat distribution.

Can double integrals be calculated without iterated integrals?

For simple regions, yes. For more complex regions, iterated integrals are the standard method.