Limits on triple integrals

As you can probably guess, there's no reason to restrict ourselves to three dimensional regions with constant boundaries. Any time we have an iterated integral (and we can go beyond triple integrals) we can have the limits be non-constant functions. There are rules, though.

1.

The limits of the inner most integral can be functions of all the other variables. Thus, if we integrate x, then y, then z, the limits on the x integral can be functions of y and z.

2.

The middle integral can have limits which depend only on the variable in the outer integral. Continuing the example in (1), this means that the limits on the y integral can be functions of z only.

3.

The outer most integral can only have constant limits of integration.

If you want to change the order of integration, you must draw the region in order to get the limits correct. If an integral has limits which violate these rules, it will be impossible to draw the region. Thus, the most general looking integral where we integrate x then y then z is of the form

$$\int_a^b \int_{f(z)}^{g(z)} \int_{h(y,z)}^{m(y,z)} \delta (x,y,z) dx dy dz.$$