The Timpson Lyceum · Real Analysis (2020)

Lecture 18 (Real Analysis 2020)

Overview

This lecture develops the central tool for deciding when a bounded function can be integrated: the Archimedes–Riemann theorem. It characterizes integrability in terms of a sequence of partitions whose upper and lower Darboux sums close the gap to zero, and uses this to prove that every monotonic function on a closed interval is integrable. Along the way it introduces regular partitions and the gap (mesh) of a partition, and closes with a fully worked computation of $\int_0^1 x^2\,dx$ from the definition. The recurring strategy — show the difference of upper and lower sums tends to zero — is the workhorse for the rest of the chapter. A short final discussion connects this picture to Leibniz notation and the way integrals are introduced in a standard calculus course.

Key Concepts

Setup and the Darboux integral. Throughout, $f:[a,b]\to\mathbb{R}$ is bounded (a necessary hypothesis — an unbounded function cannot enclose a finite area). The lower integral $\underline{\int_a^b} f$ is the supremum (least upper bound) of all lower Darboux sums; the upper integral $\overline{\int_a^b} f$ is the infimum (greatest lower bound) of all upper Darboux sums. The function is integrable exactly when these agree, and the common value is the integral:

The sandwiching inequality. For any partitions, the lower sum, lower integral, upper integral, and upper sum are ordered:

A useful rearrangement: $U(f,Q) - L(f,P) \ge \overline{\int} f - \underline{\int} f \ge 0$. Because both differences are nonnegative, driving the outer difference to zero forces the inner one to zero as well — this is why the theorem below works.

Archimedes–Riemann theorem. A bounded $f:[a,b]\to\mathbb{R}$ is integrable if and only if there exists a sequence of partitions $(P_n)$ with

Such a sequence is called an Archimedean sequence of partitions. Moreover, when it exists, $\lim_{n\to\infty} L(f,P_n) = \underline{\int} f$ and $\lim_{n\to\infty} U(f,P_n) = \overline{\int} f$. Note carefully: the theorem certifies that an integral exists; it does not by itself compute its value.

Regular partition. Splitting $[a,b]$ into $n$ equal pieces. The $i$-th partition point is

so each subinterval has the same width $\dfrac{b-a}{n}$. Regular partitions are the easiest to compute with by hand, but the theorem does not require them.

Gap (mesh) of a partition. The gap of $P$ is the length of its largest subinterval:

It is a number, not an interval. Any sequence of partitions whose gap tends to $0$ — even an irregular one built by repeatedly bisecting subintervals — qualifies as Archimedean.

Monotonic functions are integrable. Every monotonic (e.g. increasing) function on a closed interval $[a,b]$ is bounded (between $f(a)$ and $f(b)$) and integrable. Using a regular partition, the telescoping of the column heights gives the clean estimate

which $\to 0$ as $n\to\infty$. Geometrically, the "leftover" rectangles stack into a single box of width $\tfrac{b-a}{n}$ and height $f(b)-f(a)$.

Leibniz notation. Writing $\int_a^b f(x)\,dx$ records the variable of integration explicitly. Thinking of $dx$ as an infinitesimal width and $f(x)$ as a height, the integral is the limit of $\sum f(x_i)\,\Delta x$: as $\Delta x \to 0$ the deltas become $d$'s and the sum becomes $\int$. This makes Leibniz notation natural and flexible (it lets you integrate or differentiate "with respect to" either variable).

Worked Example

Prove every monotonically increasing $f:[a,b]\to\mathbb{R}$ is integrable. Take regular partitions $P_n$. On each subinterval $f$ attains its max at the right endpoint ($M_i$) and its min at the left endpoint ($m_i$), and every width equals $\tfrac{b-a}{n}$:

By the Archimedes–Riemann theorem, $f$ is integrable. $\blacksquare$

Compute $\int_0^1 x^2\,dx$ from the definition using the upper sum on a regular partition ($a=0,\ b=1$, width $\tfrac1n$, right endpoints $x_i = \tfrac{i}{n}$):

Summary

• A bounded $f$ is integrable iff its upper and lower integrals coincide; equivalently (Archimedes–Riemann), iff some sequence of partitions makes $U(f,P_n) - L(f,P_n) \to 0$.
• The standard proof technique throughout integration theory is to squeeze the difference of upper and lower sums to zero; the sandwiching inequality $U - L \ge \overline{\int} - \underline{\int} \ge 0$ is what makes this work.
• A regular partition divides $[a,b]$ into $n$ equal pieces with points $x_i = a + i\tfrac{b-a}{n}$ and uniform width $\tfrac{b-a}{n}$; the gap is the length of the largest subinterval, and any sequence with gap $\to 0$ is Archimedean.
• Every monotonic function on $[a,b]$ is integrable, because telescoping gives $U(f,P_n)-L(f,P_n) = \tfrac{b-a}{n}\bigl(f(b)-f(a)\bigr) \to 0$.
• The theorem guarantees existence of the integral but not its value; computing $\int_0^1 x^2\,dx = \tfrac13$ requires evaluating the limit of a Riemann sum directly, using $\sum_{i=1}^n i^2 = \tfrac{n(n+1)(2n+1)}{6}$.
• Leibniz notation $\int_a^b f(x)\,dx$ arises by letting the widths $\Delta x \to 0$ in $\sum f(x_i)\Delta x$, turning the discrete sum into a continuous integral.