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