How do you use the midpoint rule to estimate area?
show below please.
A good way to approximate areas with rectangles is to make each rectangle cross the curve at the midpoint of that rectangles top side.
A midpoint sum is a much better estimate of area than either a left-rectangle or right-rectangle sum.
The figure below shows you why it is better:
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.
You can approximate the exact area under a curve between a and b
with a sum of midpoint rectangles given by the following formula.
In general, the more rectangles, the better the estimate:
Where, n is the number of rectangles
is the width of each rectangle, and the function values are the heights of the rectangles.