Multiple Integrals

Multiple integrals are easy to evaluate in Mathematica, although the syntax takes a while to get used. In setting up a multiple integral remember that Mathematica integrates the outermost variable first and works it way through to the innermost variable. This is the reverse of the standard integral notation.

Example:

To integrate x^2 + y^2 on the triangular region

0 <= y <= x;  0 <= x <= 1;

we integrate the y variable first from 0 to x and then the x variable from 0 to 1. Since Mathematica evaualtes the variables from right to left we would set this up as follows:

Integrate[ x^2 + y^2, {x, 0, 1}, {y, 0, x} ]

If you would like to see some of the intermediate steps you could do the following:

Integrate[ x^2 + y^2, y ]

Integrate[ x^2 + y^2, {y, 0, x} ]

Integrate[ 4x^3/3, {x, 0, 1} ]

Example

Many integrals cannot be evaluated symbolically because there is no way to express the antiderivative in terms of familiar functions like polynomials, exponentials, logs, and trig functions. It is still quite easy for Mathematica to evaulate such integrals numerically.

NIntegrate[ E^(Cos[x y]), {x, 0, 1}, {y, 0, x} ]

Symbolic integration fails:

Integrate[ E^(Cos[x y]), {x, 0, 1}, {y, 0, x} ]