Introduction to Analysis · Chapter 08

The Riemann integral

Area defined by squeezing: partitions, upper and lower sums, the criterion that decides integrability, which functions pass it, and the theorem that ties the integral to the derivative.

a \int b f (x) d x

i \sum f (ξ_{i}) Δ x_{i}

L (f, P) \leq U (f, P)

Δ x_{i}

a \int b 1 d x = b - a

F (b) - F (a)

∥ P ∥

ξ_{i} \in [x_{i - 1}, x_{i}]

Lessons

Build the foundation

01Partitions, upper and lower sumsTo measure the area under a graph, cut the interval into pieces and on each piece take a rectangle that is certainly too tall and one that is certainly too short. Adding them gives a number above the answer and a number below it. Refining the cut can only lower the first and raise the second, so the two families of numbers press towards each other — and whether they meet is the question the whole chapter is about.Read lesson →
02Upper and lower integrals, and integrabilityThe upper sums are bounded below by every lower sum, so they have an infimum; the lower sums are bounded above, so they have a supremum. Those two numbers are the upper and lower integrals, and they always exist for a bounded function. The function is called integrable when they are equal, and the common value is the integral. Nothing here assumes they are equal, and for some functions they are not.Read lesson →
03The Riemann criterionComparing two suprema taken over all partitions is a clumsy way to decide integrability. The criterion replaces it by something a proof can actually use: the function is integrable exactly when some partition makes the gap between its upper and lower sums smaller than any given tolerance. One partition per tolerance, which is the same shape as every other definition in this course.Read lesson →
04Continuous and monotone functions are integrableTwo large families pass the criterion, by two different routes. A continuous function on a closed bounded interval is uniformly continuous, so a single fineness makes the oscillation small on every piece at once — and that is exactly the hypothesis Heine–Cantor was proved for. A monotone function may jump, and its oscillations need not be small; instead they telescope, so their total is the whole rise of the function, and the widths can be made small.

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05Properties of the integralThe integral adds, scales, splits at an interior point and respects order. None of these is obvious from the definition, because the definition is a pair of suprema over all partitions and those do not combine simply. Each is proved by choosing partitions for the pieces, refining to a common one, and applying the criterion.Read lesson →

06Riemann sumsUpper and lower sums are built from suprema and infima, which are convenient for proofs and impossible to compute. A Riemann sum instead evaluates the function at one chosen point of each piece — any point at all — and weights it by the width. For an integrable function these sums converge to the integral as the partition gets fine, whatever points are chosen, which is what makes numerical integration possible and what most calculus courses take as the definition.Read lesson →

07The Fundamental Theorem: accumulatingFix the left endpoint and let the right one move: the integral becomes a function of where you stop. That accumulating function is always continuous, whatever the integrand does, and wherever the integrand is continuous the accumulating function is differentiable with the integrand as its derivative. So integration undoes differentiation, and every continuous function has an antiderivative — which is not obvious and is the first half of the Fundamental Theorem.Read lesson →

08The Fundamental Theorem: evaluatingThe second half turns the theorem into a method. If any antiderivative of the integrand can be found, the integral is the difference of its values at the two ends, and no partition is ever formed. The proof is the Mean Value Theorem applied on each piece of a partition, and it needs no continuity of the integrand at all — only that the integrand is integrable and is the derivative of something.Read lesson →

09The integral test for seriesThe fifth chapter stated that the series of reciprocal powers converges exactly when the exponent exceeds one, proved the case of exponent one by grouping, and left the rest. The debt is paid here. A decreasing positive function and the series of its values at the whole numbers bound each other by their bars against its area, so the series and the integral converge or diverge together.Read lesson →

10Chapter practiceThis lesson mixes the chapter. The tools are the criterion, the two families of integrable functions, the four properties, and the two halves of the Fundamental Theorem — and deciding which applies is part of the work.Read lesson →

Mapping the lesson.

The Riemann integral

Build the foundation