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 this post, I’d like to take a closer look at the indefinite integral , which is closely related to the area under the bell curve . This integral cannot be computed using elementary functions. However, using integration by parts, there are some related integrals that can be computed:
Based on these examples, it stands to reason that, if can be written in terms of elementary functions, it should have the form
where is some polynomial to be determined. We will now show that this is impossible.
Suppose , a polynomial of degree to be determined. Then we have
In other words, all terms on the left-hand side except the constant term must cancel. However, this is impossible: is a polynomial of degree while is a polynomial of degree . Therefore, the left hand side must have degree and therefore cannot be a constant.
A similar argument shows that cannot have the form , where the exponents may or may not be integers.
This may be enough to convince a calculus student that there is no elementary antiderivative of . Indeed, although the proof goes well beyond first-year calculus, there is a theorem that says that if can be expressed in terms of elementary functions, then the antiderivative must have the form . So the guess above actually can be rigorously justified. References:
- Elena Anne Marchisotto and Gholam-Ali Zakeri, “An Invitation to Integration in Finite Terms,” The College Mathematics Journal , Sep., 1994, Vol. 25, No. 4 (Sep., 1994), pp. 295-308
- J. F. Ritt, Integration in Finite Terms: Liouville’s Theory of Elementary Methods, Columbia University Press, New York, 1948