Multiple Integrals in Cartesian Coordinates

.

When   is positive, this is the volume under the graph as shown for a rectangular x-y-domain:

If we group the approximating sum by fixing   and summing across   values, at each   we are approximating the volume with a slab of thickness   and the area under the curve   where   with   fixed.

The "slab" approximation replaces   with  in the sum,

and the outer sum is also approximately an integral,

This is the derivation of why the double integral is given by iterating one dimensional integrals, but the idea of the thin slabs helps us set up the more difficult problem of finding the variable limits of integration for non-rectangular domains.

Load the Animation 1 showing basic approximation for volume of the region under a positive graph

Load the Animation 2  showing the difference between step approximations and a slab in one variable

Load the Animation 3  showing the domain slice and the volume under the graph