I do not get the meaning of f(xi)+f(xi+1) in this equation. I understand that xi+1 - xi is to find the width of the interval, and the later part is trying to find the height. However, the sum of heights between the start x and end x will make the area even larger, how this helps us to find the area bounded by two x?

I do not get the meaning of f(xi)+f(xi+1) in this equation. I understand that xi+1 - xi is to find the width of the interval, and the later part is trying to find the height. However, the sum of heights between the start x and end x will make the area even larger, how this helps us to find the area bounded by two x?