# Integration of multi-valued function

Graph

I have a set of discrete (x,y) points and need to find the area between the curve and the y-axis. However, the curve seems to be a multi-valued function (multiple x for the same y). Is such an operation valid, and if so, how?

edit: The image of the curve is in the link above. Kindly take a look.

I tried to do trapezoidal integration for this in MATLAB and got a negative answer. What is the interpretation of this negative area (even when all data are positive)? To check if area becoming negative was wrong, it tried doing the integration as ($$xy - \int ydx$$). I got the same answer in both cases.

This shows the area that I need.

This shows the area that I need when multiple x occurs for a single y. Area for all x is required.

• Comments are not for extended discussion; this conversation has been moved to chat. – Aloizio Macedo Jul 12 at 22:15