The integral

has an exact answer, viz., sinc(pfT). As T®¥ the sinc function tends to zero. Divide the region from -T/2 to T/2 into N equal parts and sum the rectangles on bases Δt =T/N , showing that

Observe that this function is periodic. For a given choice of f and T construct a table to show by how much the discrete version of the integral differs from the true value. Does inclusion of only half the areas of the end rectangles make much difference? Note that you have included N+1 points in your sum. What happens if you have only N points, i.e., if you do not include one of the end rectangles. A geometric construction may assist.