# Trapezoidal Method

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?