Order of integration in triple integral Is there any hard and fast rule for what order you integrate for triple integrals. I know of Fubini's theorem but surely this doesn't cover all cases of triple integrals.
Say for example I have,
$$\int_{0}^{1} \int_{0}^{1-r^{2}} \int_{0}^{2 \pi} r^{3} d\theta dz dr $$
Why is it that I can integrate in this order as the first limit's are not a function of one of the variables the second are a function of $r$ and the last of no variable again, how would I ever know that this is the order I can integrate in apart from just inspecting.

For the integral $\int_0^{2\pi}d\theta$, it is completely independent, as you said, from the other variables, so you can evaluate it at any time and multiply the resulting double integral by its results.
For the integral $\int_0^{1-r^2}dz$, although the integrand is just $1$, the limits on the integral depend on the other variables, so after you evaluate the integral, the result will be in terms of $r$. That means that no matter what, $\int_0^1dr$ must be evaluated AFTER (on the outside) of $\int_0^{1-r^2}dz$. And those are the only real restrictions on this specific example.
In general, for most functions you will ever integrate in a multivariable calculus class, the $d\theta$ integral will be the outermost one, because it rarely (if ever) has bounds that depends on $z$ or $r$ in cylindrical coordinates or $\rho$ or $\phi$ in the case of spherical coordinates. As you said, you must inspect the integral before you start to determine if the order makes sense.
