Multiple integral

Volume under a surface - this is what a multiple integral for a function with two variables typically looks like.

In calculus and mathematical analysis, an integral is a way to calculate the limiting value a function with one variable will tend towards. Graphically, this can then be shown as the area under the graph of the function. In multivariate calculus, It is also possible to calculate integrals for functions with more than one variable. These integrals are commonly referred to as multiple integrals.

An integral for a function with two variables over a two-dimensional region, called double integral, can be shown as a surface in three dimensional space. Similarly, an integral for a function with three variables over a three-dimensional region, called triple integral, can be thought of as a hypervolume.

A double integral can also be used to define an integral over an arbitrary surface (surface integral), just like a triple integral can be used to define an integral over an arbitrary volume (volume integral).^[1]^[2]^[3]

Multiple Integral Media

Integral as area between two curves.
Transformation from cartesian to polar coordinates.
Example of a domain transformation from cartesian to polar.
A diagram of the standrad cylindrical coordinate system, with a radius, ρ, azimuth, φ and height, z. This is the ISO 31-11 notation.
A diagram of spherical coordinates, defining a point by colatitude, φ, longitude, θ, and radius, ρ.
Example: double integral over the normal region D
Example of domain in R3 that is normal with respect to the xy-plane.
Example of an improper domain.