The integral $\int_0^{\infty} \int_{x}^{\infty} f(x,y) dy dx$ is evaluated as shown in the figure below.
The integral $\displaystyle \int_{x}^{\infty} f(x,y) dy$ is first evaluated for a $dx$ red strip by letting $y$ go from $x$ to $+\infty$ and then the red strip is moved from $x=0$ to $x=\infty$. By doing this we have evaluated the integral $\displaystyle \int \int_{\Omega} f(x,y) dA$ where $\Omega$ is the blue region.
The second integral $\int_0^{\infty} \int_{0}^{y} f(x,y) dx dy$ is evaluated as shown in the figure below.
The integral $\displaystyle \int_{0}^{y} f(x,y) dx$ is first evaluated for a $dy$ red strip by letting $x$ go from $0$ to $y$ and then the red strip is moved from $y=0$ to $y=\infty$. By doing this we have again evaluated the integral $\displaystyle \int \int_{\Omega} f(x,y) dA$ where $\Omega$ is the blue region.
Both the integrals must be the same since both evaluate the double integral over the blue area, $\Omega$.