Surface integrals

I am having problems understanding the limits on a surface integral when it is over an non rectangular surface.
As an example the limits for a triangle whose corners are the points (0,0), (0,1), (1,0). How would you decide the limits?

You have variable limits. So in this case, x runs from 0 to 1, y runs from 0 to 1-x. So the integral would be like:

$\displaystyle \int_0^1 \int_0^{1-x} f(x,y) dy dx$

Note that the variable limits are on the inner integral. Otherwise it doesn't make sense.