Order of integration can be swapped if limits are constants, right?

Question

The order of integration can be easily swapped if the limits are constants, right? $$\int_{a}^{b}\int_{c}^{d}f(x,y)dydx=\int_{c}^{d}\int_{a}^{b}f(x,y)dxdy$$ It only gets computationally hard if the limits are functions of each other, right? $$\int_{a}^{b}\int_{c(x)}^{d(x)}f(x,y)dydx \neq \int_{c(x)}^{d(x)}\int_{a}^{b}f(x,y)dxdy$$ Sorry for the potentially trivial answer. Just doing a reality check over here.

Answer 1

This is true in a very general sense, by Fubini's theorem. In $\mathbb{R}^2$, the conditions for Fubini's theorem boil down to:

The integral of the absolute value over the specified range must be finite, in any one of the three senses that:

• we integrate with respect to $x$ and then $y$
• we integrate with respect to $y$ and then $x$
• we integrate simultaneously over the entire surface

If this holds, then the values of all three of the above integrals are equal.

This is usually true over a compact (i.e. bounded and closed) range, because we're usually integrating continuous functions over those ranges, and continuous functions on compact sets are necessarily bounded in modulus.