misunderstood in Fubini's theorem

Question

This question seems a little bit silly, but I have to ask. In the definition of the Fubini's theorem, these equalities bellow are regular multiplications or are iterated integrals, i.e., one integral is inside another?

I'm a little confused

Fubini's Theorem: If $f:A_1\times A_2\to \mathbb R$ is integrable then

$$\int_{A_1\times A_2}f(x,y)dxdy=\int_{A_1}dx\bigg(\int_{A_2}f(x,y)dy\bigg)=\int_{A_2}dy\bigg(\int_{A_1}f(x,y)dx\bigg)$$

I thought it first it was an iterated integral, but this example blowed my mind:

Answer 1

The second and third are iterated integrals. The first is a single integral, which in simple cases, can still be calculated "directly" (consider the integral of 1 over [0,1] x [0,1]). The power of Fubini's theorem is that, in the many cases where calculating an integral in more than one variable is too difficult, you can simply resort to an iterated one to get the right value.

EDIT: to address your edit, the first equality is something you get by transitioning into an iterated integral from the noniterated(?) integral. Then the second integral can be written $$\int_{-n}^{n}dy\int_{-n}^{n}e^{-(x^2+y^2)}dx=\int_{-n}^{n}e^{-y^2}dy \int_{-n}^{n} e^{-x^2} dx$$ since with respect to integration over $x$, the $e^{-y^2}$ is a constant, so we can pull that factor out. Then we simply observe that the two integrals on the right are the same expression, just with different "dummy" variables. So we relabel both $x$ and $y$ with $t$.

Check out https://en.wikipedia.org/wiki/Gaussian_integral#Computation for a simple but important computation that has many consequences for many fields of math.