TeXipedia

iint

Represents a double integral in multivariable calculus, used for integrating functions over two-dimensional regions.

Overview

Essential for calculating quantities across two-dimensional domains, particularly in physics, engineering, and advanced mathematics.

  • Commonly used to find areas, volumes, and surface integrals
  • Appears frequently in electromagnetic theory for surface flux calculations
  • Key notation in vector calculus for computing multiple integrals over regions in the plane
  • Often paired with boundary specifications to define the region of integration

Examples

Double integral over a region R in the xy-plane.

∬Rf(x,y) dx dy\iint_R f(x,y)\,dx\,dy 
\iint_R f(x,y)\,dx\,dy

Computing the mass of a lamina with density function ρ(x,y).

M=∬Dρ(x,y) dx dyM = \iint_D \rho(x,y)\,dx\,dy 
M = \iint_D \rho(x,y)\,dx\,dy

Calculating electric flux through a surface S.

ΦE=∬SE⃗⋅dS⃗\Phi_E = \iint_S \vec{E} \cdot d\vec{S} 
\Phi_E = \iint_S \vec{E} \cdot d\vec{S}