I have the joint density function:
First, I had to find k. So I integrated for x with given limits and then for y with the given limits and ended up with k=1/4. Correct?
Now I have to find the joint cumulative distribution function... stuck. Do I just put in k=1/4 into the density function and then find the derivative?
The joint cdf is defined as , so just use the joint pdf to calculate that probability like you would any other (e.g. if you were calculating what would you do? Extend this method to arbitrary x and y, and you will have the joint cdf). The joint cdf that you posted is correct if you assume that , but that might not be the case. The cdf must be defined for everything in the plane.
The marginal densities are wrong. You integrate the unwanted variable out using the joint pdf, not the joint cdf.