The joint density of $X$ and $Y$ is given by $$f_{X,Y}(x,y)= \left\{\begin{matrix}(xy), \mbox{ } 0\leq x \leq1 , 0\leq y \leq 2 \\ 0, \mbox{ otherwise}\end{matrix}\right.$$
Evaluate $P(X+Y\leq1)$.
I'm not sure if I'm doing this correctly. Please let me know if something is off in my solution. Thanks for reading!!
My solution:
$$ P(X+Y\leq1)= \int_{0}^{1}\int_{0}^{1-x}(xy)dydx = \int_{0}^{1}\left [ \frac{xy^{2}}{2} \right ]\Big|_0^{1-x}dx= \int_{0}^{1}\left [ \frac{x(1-x)^{2}}{2} \right ]dx$$ $$=\frac{1}{2}\int_{0}^{1}(x-2x^{2}+x^{3})dx=\frac{1}{2}\left [ \frac{x^{2}}{2}-\frac{2x^{3}}{3}+\frac{x^{4}}{4} \right ]\Big|_0^1$$ $$=\frac{1}{2}\left [ \frac{1}{2}-\frac{2}{3}+\frac{1}{4} \right ]=\frac{1}{2}\left [ \frac{6-8+3}{12}\right ]=\frac{1}{24}.$$