If you were to do a bit of experimentation, you would see that using rectangles
is not always the most effective way to approximate the area under a curve.
One simple improvement that we can make to obtain better approximations is to
estimate the area under a curve by using a series of *trapezoids* instead
of rectangles. See this for an illustration.

As before, we divide the interval (*a*, *b*) into *N* equal subintervals, each of
size

Also as before the endpoints of the ith subinterval are *a*+(*i*-1)*h* and *a*+*ih*.
The (x,y)-coordinates of the four corners of the trapezoid that we use to
approximate the area under the curve in this subinterval are then

- (
*a*+(*i*-1)*h*, 0) (the lower-left corner) - (
*a*+(*i*-1)*h*,*f*(*a*+(*i*-1)*h*)) (the upper-left corner) - (
*a*+*ih*, 0) (the lower-right corner) - (
*a*+*ih*,*f*(*a*+*ih*)) (the upper-right corner)

The area of this trapezoid is given by the length of the base times the average of the heights of the two sides:

Now to approximate the area under the curve, we simply add up the areas of all of the trapezoids:

If you think about what is going on in this summation, you can come up with
a new summation that can be computed more efficiently. Except for *f*(*a*) and
*f*(*b*), every other value of *f* that is computed in this summation is computed
*twice*. Why is that?

Based upon this realization, it is not difficult to come up with a summation that requires half as much work to compute:

Hamlet Project

Department of Computer Science

University of Utah