Here's the problem description:
Let the function $f(x,y)$ be defined by
$
f(x)=
\begin{cases}
g(x)/x,&\text{if}\, 0 < y < x\\
0, &\text{otherwise}
\end{cases}
$
where $g(x)$ is a non-negative function defined on $(0, \infty)$ and $\int_{0}^{\infty}$ $g(x) dx = 1$.
Compute
$\int_{-\infty}^{\infty}$ $\int_{-\infty}^{\infty}$ $f(x,y)dx dy$
I swear this should be pretty easy but for some reason I'm just stumped. I tried integration first with respect to $dx$ and splitting up the improper integral on the inside from $-\infty$ to $0$ and $0$ to $\infty$… but I just ended up with something that I couldn't really compute.
I'm also not really sure how I'm supposed to use the fact that $\int_{0}^{\infty}$ $g(x) dx = 1$ when I have to integrate $g(x)/x$. I'm guessing I'm pretty rusty on my calc.
I would appreciate it if anyone could help me out here.
Best Answer
HINT: Observe that $$ \int_0^x (g(x)/x)\,dy=(g(x)/x)\cdot1\Bigr|_0^x=g(x). $$ Let $D=\{(x,y):0\leq x\leq y\}$. Then $$ \int\int_{D}f(x,y)\,dx\,dy=\int_0^\infty g(x)\,dx=1. $$