Confirming the domain for this integral

1. Sep 24, 2013

fateswarm

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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).