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I'm working on a problem where $f(x,y)=c(x+y)$ is a joint PDF where $0

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The region $0

If for example you want to find $c$, you could try the integral $$\int_{y=0}^1 \int_{x=0}^y c(x+y) \; dx \; dy$$ and set that equal to $1$ to solve for $c$, though there are other possible approaches.