Order of integration in triple integral

Question

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.

Answer 1

$\theta$ is independent of $r$ and $z$ so the corresponding integral $\int_0^{2\pi}\dots d\theta$ can be at any of positions (1st, 2nd or 3rd).

A bound for $z$ depends on $r$ thus first has to be computed the integral $dz$, only then the integral $dr.$

Convenient possibilities giving the same result without any further transformation are $$\int_{0}^{2\pi} \int_{0}^{1} \int_{0}^{1-r^{2}} r^{3} dz\; dr\; d\theta = \int_{0}^{1} \int_{0}^{2 \pi} \int_{0}^{1-r^{2}} r^{3} dz\; d\theta \;dr=\int_{0}^{1} \int_{0}^{1-r^{2}} \int_{0}^{2 \pi} r^{3} d\theta\; dz\; dr.$$

Answer 2

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.