bound of marginal distribution with indicator

Question

I'm trying to get the marginal density function of $X$ and $Y$ where I'm given the joint density as $$f_{X,Y}=8xy\;I_{(0,y]}(x)I_{(0,1)}(y)$$ It is easy to see that the marginal of $X$ is $$f_X=\int_0^18xy \;dy=4x$$ and the marginal of $Y$ is $$f_Y=\int_0^y8xy\;dx=4y^3$$ but what about their boundaries? Intuitively I think the bound on $Y$ is $(0,0)$ since it is as the indicator indicates; however I have no idea what bound I should give to $X$. the indicator on $X$ is $(0,y]$ and that bound does not make sense on a marginal of $X$, please show me what I should do, any help is appreciated!