please look at the following question,
Let $X$ denote the diameter of an armored electric cable and $Y$ denote the diameter of the ceramic mold that makes the cable. Both $X$ and $Y$ are scaled so that they range between $0$ and $1$. Suppose that $X$ and $Y$ have the joint density
$f(x, y) =\begin{cases} \frac1y,&0
when i solved it i got the following limits of x and y, 0->1/2 and 0->(1/2-x) however according to the book the correct limits are 0->1/4 and x->1/2-x I am just confused how to plot this function in order to find out where that 1/4 came from ?
According to the book the solution is,
$\begin{align*} P\left(X+Y > \frac12\right)&=1-P\left(X+Y < \frac12\right)\\ &=1-\int_0^{1/4}\int_x^{1/2-x}\frac1y dy\,dx\\ &= 1-\int_0^{1/4}\left[\ln\left(\frac12-x\right)-\ln x\right]dx\\ &=1+\left.\left[\left(\frac12-x\right)\ln\left(\frac12-x\right)-x\ln x\right]\right\vert_0^{1/4}\\ &=1+\frac14\ln\left(\frac14\right)\\ &=0.6534. \end{align*}$
please guide me where that 1/4 came from and how can i plot that question to clearly understand that ?