1. ## Limits of Integration

Joint pdf given as kxy for 0 < x < y < 1.

Find the value of k.

I understand the process of finding k - doing the double integral and setting it to 1. What I dont understand is the limits of integration for y.

I've seen two different limits set, but I still cannot seem to figure out how and why it is done.

I have seen the integral of x from 0 to 1 and the integral of y from x to 1. I have also seen the integral of x from 0 to 1 and the integral of y from 0 to y. Both give the correct answer of k = 8. My question is how do you go about choosing the limits for the y integral?

Thanks a lot!

2. Originally Posted by scothoward

I have seen the integral of x from 0 to 1 and the integral of y from x to 1. I have also seen the integral of x from 0 to 1 and the integral of y from 0 to y. Both give the correct answer of k = 8.
Of course, it's just reversing integration order.

Originally Posted by scothoward

My question is how do you go about choosing the limits for the y integral?
You mean how you get new bounds when reversing integration order?

3. Actually, I just dont understand the concept of how the bounds of y are chosen either as x to 1, or 1 to y

Now that I think about it a bit more...would x to 1 be because y > x, so the lowest y can go is x?? If that is correct, what is the rationale for 1 to y??

Thanks again

4. Well, given $\displaystyle 0\le x\le y\le 1,$ bounds of the outter integral should be integers, that's why $\displaystyle 0\le x\le1;$ as for bounds of the inner integral, it's just $\displaystyle x\le y\le1.$