**Multi-variable Integration**

The Integral commands in EES can be nested or serially-related to provide multiple variable integration. EES evaluates the integral numerically and so the calculation effort increases significantly as the number of variables is increased. An example that uses double integration is provided below.

"This integral can be evaluated analytically and the exact result is 104.1428. In this case, the inner integral (integration variable y) uses a fixed step size of 0.05 and the outer integral (integration variable x) uses a fixed step of 0.03 (100 steps).

F=**integral**(integral(xy,y,0,x,0.05),x,0,3,0.03)

xy=x^3*y^2

The result for F is 104.2.

Note that this function could have been equivalently entered as follows:

xy=x^3*y^2

F1=**integral**(xy,y,0,x,0.05)

F2=**integral**(F1,x,0,3,0.03)

F2 will have the same value as F.

You may wish to investigate the effect the step size has on the calculation effort and accuracy of integration.

Note: EES cannot always distinguish the difference between the need to do multi-variable integration or sequentially separate single-variable integration. Multi-variable integration is much slower than single-variable integration. If EES is attempting to use multi-variable integration when you wish to have single variable integration, place one or more of the integrals in separate Subprograms.