Probability integration question

[a_man]

Homework Statement

Let X denote the diameter of an armored electric cable and Y denote the diameter of the ceramic mold that makes the cable. Both X and Y are scaled so that they range between 0 and 1 . Suppose that X and Y have the joint density

f(x,y) = 1/y, 0<x<y<1
f(x,y) = 0, elsewhere

Find P(X + Y > 1/2)

Homework Equations

The Attempt at a Solution

Do not I just need to integrate 1/y from x to 1 with dx then from there integrate one more time with from 0 to 1/2 with dy ?

I am really confused with this question because if we integrate 1/y we get ln (y) and if we integrate one more time, we get y *(ln (y) - 1) but we cannot have ln (0).

Some please help me. This question is bothering me for the whole day.
BTW: I am sorry that I do not know how to put the integration symbol yet.

 

[LeonhardEuler]

It would probably be helpful to draw a diagram so that you can see the area where you need to integrate. You are looking for X+Y>1/2. Graph the line X+Y=1/2. This is the boundary of the constraint, and on one side of the line it holds, on the other it fails. Also, the function is nonzero only for X<Y, so this is another constraint on the area of integration. If you draw a picture, it should be clear what region to integrate in and how to set the limits.

As far as your second point about ln(y) for y=0, remember that it is y*ln(y). The limit has a definite value. (hint: L'Hopital's rule)

And about drawing integrals, hit the quote button to see how I did this:
\int_{x=0}^{x=1}f(x)dx