How do I determine the exact domain of y given a graph?

Question

I've made a post earlier about finding the joint PDF of a given figure:

$$f_{X,Y}(x,y) = \begin{cases} 1\over 2& ,0\le x \le 2 , \max(0, x-1) \le y \le \max(1, x) \\ 0 & \text{,otherwise}\end{cases}$$

Someone told me how to find it using the area of the figure but I have one more issue. If I want to find the marginal PDF of X, I need to know the exact domain of y, in order to be able to integrate how do I that? I can't integrate $$\int_{max(0, x-1)}^{max(1,x)} 1/2dy$$ because I don't know the exact values. How can I find them?

Answer 1

You already know the domain of $y$ when $x$ is given as they are shown in your integration limit.

\begin{align} f_X(x) &= \int_{\max(0,x-1)}^{\max(1,x)} \frac12 \, dy \\ &= \frac12 \left[ \max(1,x) - \max(0, x-1)\right]\\ &= \frac12 \end{align}