Given an empirical PDF of a continuous random variable $X$, then integrating over its entire defined domain will yield an area of size 1. To find the probability of $X \ge x_1 \land X \le x_2 $ (as in the image), we can simply integrate the PDF over that range.
Since the area represents the cumulative probability of the range it is denoted by, we can ask questions like "Given a cumulative probability of 0.1 and the mode of the PDF, what is the range of the domain given an approach where we approximate $PDF(x)$ so that it demarcates the range for a known area?".
Question: How to choose $y$ of the blue line without having to do it numerically then?
Bonus question: How to handle multi-modal cases?