It is implied that f and g are zero everywhere where they are not explicitly defined above (I didn't bother typing it out). I'm not sure if this changes anything.
I was originally trying to solve a probability problem where f and g above are probability density functions for the stochastic variables X and Y. The problem was to find the probability density function for where X and Y are independent. I figured I could find the distribution for W and then take the derivative of that.
No, w is not a variable. For the purposes of this problem, w is a constant.
When x= 1, y= w. When y= 1,
If you wish to integrate with respect to y first, for each x, y will vary from 0 to . If you wish to integrate with respect to x first, for each y, x will vary from 0 to .
Your integral is
or
I had tried both those integrals before posting the question but they both give the wrong answer. I figured out that you need to split the integral into two different areas to account for all values of w. It becomes obvious if you try to graph the area for different values of w.