You shouldn't have an x variable in the inner bounds of integration since you will end up xs when you do the last integration.
So f(x,y) = kxy, it's a joint probability density function and I must find k so that the integral over the valid range is 1.
My multivariate calculus is a little rusty and my first attempt which looked like (attached) yielded a k of -0.0012 and an integral of 1.
However I solved for k by double integrating with the calculator and setting the to 1 so of course it will yield 1 for the given interval, what I want to confirm is that the integral was setup properly.
Yes, the region is correct.
What are the bounds for y when x is a certain value?
This is the same thing as asking, what are the bounds when you take a vertical line slice of a region.
What are the bounds for y between x = 0 and x = 20?
What are the bounds for y between x = 20 and x = 30?