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$$f(x,y) = \begin{cases} \dfrac{x^2+y}{4} & 0<x<y<2 \\[8pt] 0 & \text{otherwise} \end{cases} $$
Find Marginal PDF for $x$ and $y$.
I know the formula is $\int^\infty_{-\infty} f(x,y)\,dy = $ Marginal PDF for $x$ and the same for $y$ just the integral with respect to $x$. However I don't understand what limits for integration I should use or how to go about finding them.
By the definition of $f$ one gets $$ \int^\infty_{-\infty} f(x,y)\,dy=\int^x_{-\infty} \,0\,dy+\int^2_x f(x,y)\,dy=\int^2_x f(x,y)\,dy. $$ Do you see it?
Similarly for $x$.
The pdf is non-zero if and only if $0<x<y<2$. So when you integrate with respect to $y$, the bounds are $x$ and $2$, and $0<x<2$. When you integrate with respect to $x$, the bounds are $0$ and $y$, and $0<y<2$.
