Multiple Integrals

 

Double Integral

A double integral consists of two successive integrations of a function of two variables. We calculate each integration independently of the other. 

We initially calculate the integration inside the brackets, considering x as constant and then we calculate the resulting function with respect to x. Geometrically the result represents the volume between xy plane and the surface f(x,y) in the three dimensional Cartesian System inside the limits of the integration.

Image 1: Geometric Interpretation of Double Integral

 

Example 1:

 

 

Example 2:

 

Triple Integral

Extending the double integral we get the triple integral. A triple integral consists of three successive integrations of a function of three variables. We calculate each integration independently of the other. 

 

Example:

 

 

Multiple Integrals

An n multiple integral consists of n successive integrations of a function of n variables. We calculate each integration independently of the other.