Confirming the domain for this integral

[fateswarm]

Homework Statement

In a Probabilities question it was known that 0<=x<=y<=1.

It was asked what is the P(X+Y>1).

So I had to find the domain for that integral first.

Homework Equations

The Attempt at a Solution

I started with a graph sketch for 0<x<y<1 which gave half of the 1X1 square, the north-west triangle.

Then I did a graph sketch of x+y>1 which gave the north-east trangle of the half of the 1X1 square.

Combining those I get a triangle at the top corresponding to a quarter of it all.

So, is it right to assume that it is wrong to do it directly on one integral without splitting it in two triangles?

My current solution is that the integral must by multiplied by 2 and be done in the domain of 0.5 to 1 for dy externally of the integral and inside 1-y to 0.5 for dx, or equivalently, 0 to 0.5 for dx externally, and 1-x to 1 for dy internally (each integral multiplied by 2) (plus two other solutions representing the other triangle).

 

[Nino MMP]

But to make it simpler, I think that the integral should be done in the domain of 0 to 1 for both dx and dy, but then divided by 4.Which is the correct solution?