I'm given a $$f_{X,Y}(x,y) = \begin{cases} cx, & \text{x > 0, y > 0, 1}\ \leq \ x+y \ \leq 2, \\ 0, & \text{elsewhere.} \end{cases}$$ and trying to find a the constant $c$.
I've set the x range to $$0 < x < 2$$ and y range to $$1-x < y < 2-x$$ but I'm confused as if this should be $0 < y < 2-x$ instead
The answer said $$c\int_{0}^1\int_{1-x}^{2-x}xdydx + c\int_{1}^2\int_{0}^{2-x}xdydx$$ and continue the calculation from here.
Why should the equation be formed in this way?
So you need to calculate the double integral $$ \iint_Ef(x,y)\;dxdy\quad E=\{(x,y):x>0,y>0,1\le x+y\le 2\} $$ The region $E$ is not a "normal domain" in $\mathbf{R}^2$. So in order to write the integral as iterated integrals, you need to split the region into two parts so that they are both normal domains. Your solution divide the region as follows: