All courses › Multivariable Calculus › Double integral

Double integral

A double integral sums a function over an area, and is usually computed as two ordinary integrals in a row. Fubini's theorem says the order of integration does not matter for continuous functions on a rectangle.

∬Rf(x,y) dA=∫ ⁣ ⁣∫f(x,y) dx dy\iint_R f(x,y)\,dA = \int\!\!\int f(x,y)\,dx\,dydouble integral, Fubini's theorem

Symbols

dAdAarea elementm²

Example

∬[0,1]×[0,1]xy dA=∫01x dx⋅∫01y dy=12⋅12=0.25\displaystyle\iint_{[0,1]\times[0,1]} xy\,dA = \int_0^1 x\,dx\cdot\int_0^1 y\,dy = \tfrac12\cdot\tfrac12 = 0.25.

When the integrand is a product f(x)g(y)f(x)g(y) on a rectangle, the integral splits into two simple integrals.
Practise multiple integrals for free →

← Lagrange multipliers · Polar coordinates →

Part of Multivariable Calculus: Multiple integrals.