Joint Distribution

#1
Hi everyone,
I've got a problem with a joint probability density function question.

Given f(x, y) = e^-(x+y),
0 <= x < ∞ ,
0 <= y < ∞ ,
and 0 otherwise
I need to verify that f is a valid joint PDF.

The fact that f(x, y) >= 0 for all x, y is obvious. However I'm having trouble with showing:

∫ ∫ f(x,y)*dx*dy = 1

...where both integrals are from 0 to ∞.

and the last thing I can't quite figure out is how to find P(X<Y).

Any input would be greatly appreciated.
Thanks!