More Than Antiderivatives
I turn away with fright and horror from the lamentable evil of functions which do not have derivatives.
— Charles Hermite, in a letter of 1893
Ask a school student what integration is, and you will probably hear “the opposite of differentiation”. That is a wonderful fact, the fundamental theorem of calculus, and it makes most of the integrals you meet at school routine. But it is a theorem about integrals, not a definition of them, and it hides most of what makes integration interesting.
Integration is older than differentiation by about two thousand years. Archimedes found the area of a parabolic segment and the surface area of a sphere by summing thin slices, long before anyone thought of derivatives. And integration reaches places differentiation cannot. Every reasonable function can be differentiated by mechanical rules, but most functions cannot be integrated in a formula, which raises a natural question: when can they, and what do we do when they can’t? Integrals define new functions, prove inequalities between numbers, compute areas and volumes that have no other description, and give exact values of infinite sums and products. This chapter is about all of these.
What an integral really is
The chapter begins at the beginning, with the definition (Section 4.1). An integral is a limit of sums of thin rectangles; computing even \(\int_0^a x^5\,\mathrm{d}x\) from that definition, without antiderivatives, is a substantial piece of work, and it makes you appreciate the fundamental theorem. With a precise definition comes a precise question: which functions are integrable? The answer, and the problems it raised, led in the early twentieth century to Lebesgue’s theory of integration, the foundation of modern probability.
Integrals that prove things
Some of the most surprising results in this chapter use integrals to prove facts that seem to have nothing to do with areas. A single integral of a positive function equals \(\frac{22}{7} - \pi\), which proves \(\pi < \frac{22}{7}\) in one line (Section 4.3). Powers of \(\sin t\) lead to Wallis’s infinite product for \(\pi\) and to the value \(\left(\frac12\right)! = \frac{\sqrt\pi}{2}\) (Section 4.4). The integral of \(x^x\) turns into the sum \(\sum (-1)^{n+1}n^{-n}\) (Section 4.5). And a single integral, \(\int_0^x \frac{\mathrm{d}u}{1+u^2}\), can be used to build trigonometry from scratch (Section 4.2).
What you will meet at university
The first-year Analysis course proves the fundamental theorem of calculus properly, from the definition of the integral. Later courses build in several directions. Measure theory and the Lebesgue integral make integration work smoothly with limits and underpin probability. Vector calculus integrates over curves, surfaces and volumes, and Green’s and Stokes’s theorems relate them (Section 4.9 is a taste). Complex analysis evaluates real integrals by detours through the complex plane. The calculus of variations asks which function makes an integral as small as possible (Section 4.7), and leads to the principle of least action in physics. And differential algebra explains which integrals have formulas at all (Section 4.6).
The questions that run through this chapter
What is an integral, and which functions have one? How does the definition as a limit of sums connect with antiderivatives?
What can integrals prove? How can an area tell us about \(\pi\), about factorials, about infinite sums?
What can we do when there is no formula? How do we know an integral has no formula, and what can we compute anyway?
Ideas in this chapter
| § | Topic | Idea |
|---|---|---|
| 4.1 | The definition of the integral | Lower and upper sums; telescoping |
| 4.2 | Trigonometry from \(\int \frac{\mathrm{d}u}{1+u^2}\) | Substitutions that preserve an integral |
| 4.3 | \(\frac{22}{7} > \pi\) | An inequality as an integral of something positive |
| 4.4 | Wallis and \(\left(\frac12\right)!\) | Reduction formulae, an invariant, a squeeze |
| 4.5 | \(\int_0^1 x^x\,\mathrm{d}x\) | Series expansion and term-by-term integration |
| 4.6 | Integrals with no formula | Impossibility via polynomial algebra |
| 4.7 | The calculus of variations | Completing the square in an integral |
| 4.8 | Archimedes’ hat-box theorem | Simplifying an integrand reveals geometry |
| 4.9 | Holditch’s theorem | Area from a boundary integral; elimination |
How to read this chapter
Several sections involve heavy but routine calculation: polynomial division, integration by parts, expanding products. Do these yourself, with pencil and paper, before reading the text. Then look for the idea that made the calculation work: an invariant, a symmetry, a cancellation. The calculation is what an examiner checks; the idea is what you will remember and use again.