Now, start by splitting the region you are evaluating the integral over into sub intervals, and also mark in their midpoints, such that and approximate the integral using rectangles of the same length as one of these subintervals, i.e. and width evaluated at the midpoint of each subinterval, i.e. . Then the area of each rectangle is and the total area can be approximated by .
In order to improve on this approximation, we need to increase the number of subintervals, to make the length of each subinterval smaller and to decrease the error. So the exact area is the limit of this sum as we make .
Therefore , and since we are making the subintervals extremely small, the midpoint ends up being the ONLY point between the two endpoints of each subinterval, which means we can simplify using the rearranged form of the Mean Value Theorem that is given above. So
Hope that helped