How to Approximate Area with Midpoint Rectangles A good way to
approximate areas with rectangles is to make each rectangle
cross the curve at the midpoint of that rectangle's top side. A
midpoint sum is a much better estimate of area than either a
left-rectangle or right-rectangle sum. The below figure shows
why. You can see in the figure that the part of each rectangle
that’s above the curve looks about the same size as the gap
between the rectangle and the curve. A midpoint sum produces
such a good estimate because these two errors roughly cancel out
each other. The figure above shows how you’d use three midpoint
rectangles to estimate the area under from 0 to 3. For the three
rectangles, their widths are 1 and their heights are f(0.5) =
1.25, f(1.5) = 3.25, and f(2.5) = 7.25. Area = base x height, so
add 1.25 + 3.25 + 7.25 to get the total area of 11.75. Using the
definite integral, you find that the exact area under this curve
turns out to be 12, so the error with this three-midpoint-
rectangles estimate i