Numerical integration is a standard topic in first-semester calculus. From time to time, I have received questions from students on various aspects of this topic, including:

Why is numerical integration necessary in the first place?

Where do these formulas come from (especially Simpson’s Rule)?

How can I do all of these formulas quickly?

Is there a reason why the Midpoint Rule is better than the Trapezoid Rule?

Is there a reason why both the Midpoint Rule and the Trapezoid Rule converge quadratically?

Is there a reason why Simpson’s Rule converges like the fourth power of the number of subintervals?

In this series, I hope to answer these questions. While these are standard questions in a introductory college course in numerical analysis, and full and rigorous proofs can be found on Wikipedia and Mathworld, I will approach these questions from the point of view of a bright student who is currently enrolled in calculus and hasn’t yet taken real analysis or numerical analysis.

In the previous post in this series, I discussed three different ways of numerically approximating the definite integral , the area under a curve between and .
In this series, we’ll choose equal-sized subintervals of the interval . If is the width of each subinterval so that , then the integral may be approximated as
using left endpoints,
using right endpoints, and
using the midpoints of the subintervals. We have also derived the Trapezoid Rule
and Simpson’s Rule (if is even)

.

There is a somewhat surprising connection between the last three formulas. Let’s divide the interval into subintervals with and , , , and so on. Then Simpson’s Rule becomes

.

Next, let’s divide the interval into subintervals, but let’s not redefine the values of and the . Instead, the width of each subinterval will be , which is equal to . (In other words, since there are half as many subintervals, each one is twice as long.) Also, the endpoints of these subintervals will be , , , and so on. So, keeping the same labeling convention as with Simpson’s Rule, the Trapezoid Rule becomes

.

(Again, the width of the subintervals in this case is , where .) Furthermore, the midpoint of subinterval will be , the midpoint of subinterval will be , and so on. Therefore, keeping the same labeling convention, the Midpoint Rule becomes

.

It turns out that , a certain weighted average of and , is equal to

.

So, if the Midpoint Rule and the Trapezoid Rule have already been computed for subintervals, then Simpson’s Rule for subintervals can be computed at almost no additional effort.

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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