Fubini's theorem

When we have a continuous function with more than one variable -let's call it $f(x,y)$- we can compute its integral on a region of the plane - let's call it $R$ - instead of the interval $[a,b]$ with which we are used to working.

We will then write , $\displaystyle\int_R f(x,y) \ dA$ where $R$ is the region or domain of integration and $dA$ is the 'differential unit of the area'.

This type of integrals have the following properties:

$$\displaystyle \int_R K\cdot f(x,y) \ dA=K\cdot \int_R f(x,y) \ dA$$ where $K$ is a constant.

$$\displaystyle \int_R f(x,y) \pm g(x,y) \ dA = \int_ R f(x,y) \ dA \pm \int_R g(x,y) \ dA$$

If $R=R_1\cup R_2$, so if $R$ is the disjunct union of $R_1$ and $R_2$ $$\displaystyle\int_R f(x,y) \ dA= \int_{R_1} f(x,y) \ dA + \int_{R_2} f(x,y) \ dA$$

In the case where we integrate on a rectangular region $[a,b] \times [c,d]$, we will write the integral: $$\displaystyle \int_c^d \int_a^b f(x,y) \ dxdy$$

We must take into account that, in this case, $[a,b]$ is the interval of integration in the $x$ axes, while $[c,d]$ is the interval in $y$.

In this case we can write $$\displaystyle \int_c^d\Big( \int_a^b f(x,y) \ dx\Big)dy= \int_a^b\Big( \int_c^d f(x,y) \ dy\Big)dx$$

This property is called Fubini's theorem.

In order to compute these integrals, we will first compute the inside integral by taking the other variable as a constant and then, once the first variables is 'eliminated', we integrate regarding the second one.

$$\displaystyle \int_0^1\int_0^{\frac{\pi}{2}} e^y\sin x \ dxdy=\int_0^1e^y\int_0^{\frac{\pi}{2}} \sin x \ dxdy \int e^y\Big[-\cos x\Big]_0^{\frac{\pi}{2}} \ dy=$$ $$\int_0^1 e^y \ dy= \Big[e^y \Big]_0^1e-1$$