Given the following PDF of continuous 2 random variables:

$$ f_{X,Y}(x,y)=\begin{cases} y^2 & 0\le y\le x\le 1;\newline 0 & \text{otherwise}. \end{cases} $$

Graph showing the region of integration with x and y random variables:

limit of 0<y<x<1

Question is to find the marginal CDF $F_{Y}(y)$ and $P[Y > x]$

My attempt to find joint CDF: $F_{X,Y}(x,y)$ $$ \begin{alignedat}{0} F_{X,Y}(x,y)&=\int_{-\infty}^{x}\int_{-\infty}^{x}y^2 dydx \newline &=\int_{0}^{x}\int_{0}^{x}y^2 dy dx\newline &=\int_{0}^{x}\frac{y^3}{3}\Bigg|_{0}^{x} dx=\int_{0}^{x}\frac{x^3}{3}dx\newline &=\frac{x^4}{12}\Bigg|_{0}^{x}=\frac{x^4}{12} \end{alignedat} $$

And then to find marginal CDF of: $F_{Y}(y)=F_{X,Y}(x,y)$

$$ F_{Y}(y)=F_{X,Y}(\infty,y)= $$

But where do I go from there? What does setting x to $\infty$ even mean?

And the question to find $P[Y>X]$, shouldn't it be 0 since $0\leq y\leq x\leq 1$?