Understanding Joint PDF and Independence in Probability: Solving for P(X+Y<=2)

[cjaylee]

Let the joint PDF of (X,Y) be of the form:
f(x,y) = 1/8x(x-y), 0<=x<=2, |y|<=x
f(x,y) = 0 elsewhere

Find P(X+Y<=2).

The answer that my teacher gave was

P(X+Y<=2)=∫₀¹dx ∫_-x^x 1/8x(x-y)dy + ∫₁²dx ∫_-x^2-x 1/8x(x-y)dy

I do not understand how my teacher could separate the integral like that ∫ dx ∫ dy when the function has both the variable x and y. Shouldn't that be only possible when we can separate the function into 2 separate functions that are independent of each other?

Thanks for the help!

 

[bpet]

Hint: draw a sketch of the regions of integration in (x,y).