Let X and Y be continuous random variables that have the following joint probability density function: $$ f(x,y) = \begin{cases} 3y\ ,\ 0\le x\le y \le1 \\ 0 \ \ \ ,\ elsewhere \end{cases} $$
I'm trying to find $P(Y < 1/2)$ I'm having difficulty understanding the precise steps, and because 1/2 is a constant, I'm wondering if I'm supposed to find this probability via $\int_0^{1/2}3y \ dy$, $\int_x^{1/2}\int_0^y 3y \ dy$, or some other limits and ordering here. Every resource, including online, textbook, and accompanying exercises includes another variable in the parentheses.
$f_Y(y)=\int f_{X,Y} (x,y)dx=3y^{2}$ for $0\leq y \leq 1$. Hence $P(Y<\frac 1 2)=\int_0^{\frac 1 2} 3y^{2}dy=\frac 1 8$.