By David Bressoud @dbressoud
As of 2024, new Launchings columns appear on the third Tuesday of the month.
As explained in last month's Launchings, Cauchy had realized that we cannot define integration as the inverse process of differentiation. That is because there are continuous functions such as $e^{-1/x^2}$ (defined to be 0 at $x=0$) for which there is no closed from of which it is the derivative but for which the definite integral exists. Integration must be defined as a limit of approximating sums, what today we somewhat erroneously refer to as ``Riemann'' sums. Cauchy defined his definite integrals as limits of these sums five years before Riemann was born. (How Riemann's name got attached will be explained at the end of this column.) But Cauchy now had a problem which was to explain why definite integrals could be evaluated by finding an anti-derivative.
$$The first step is to turn the definite integral into a function. This can be done by turning the upper limit into a variable,
where the lower limit, $a$, is still constant. If $f$ is continuous, then for each value of $x$ there is a well-defined value of $F$, so this really is a function. It should be noted that students often struggle with this idea of turning a definite integral into a function. Little time is spent clarifying the importance of what has just been done. Functions defined as definite integrals occur so infrequently within the calculus course that students come to expect that definite integrals always have constant limits. At least partly for this reason, Pat Thompson at Arizona State University has chosen to make definite integrals with variable upper limit the very first definite integrals his students see.
$$Since this is a function, one can seek to differentiate it.
Observe that this is valid whether $\delta$ is positive or negative. If $\delta < 0$, say $\delta = -\epsilon$, then
$$Let $M_{\delta}(x)$ be the maximum value of $f$ on the interval between $x$ and $x+\delta$, and let $m_{\delta}(x)$ be the minimum value (strictly speaking, $M_{\delta}(x)$ is the least upper bound on this interval and $m_{\delta}(x)$ is the greatest lower bound). We know that
$$(Check that these inequalities hold whether $\delta$ is positive or negative.) Furthermore, $f$ is continuous at $x$ and therefore $\lim_{\delta\to 0} \left( M_{\delta}(x) - m_{\delta}(x)\right) = 0$. It follows that
$$The conclusion is that every continuous function $f$ has an antiderivative that can always be expressed as a definite integral with a variable upper limit,
$$Notice that we did not care what value we chose for the constant $a$. The only restriction is that it must lie within the interval where $f$ is continuous. This tells us that there is not just one antiderivative, but an uncountable number of them, a different antiderivative for each constant.
$$We now can ask: If there is some nice expression $F$ for an antiderivative of $f$, $F'(x) = f(x)$, what is the relationship between $F$ and the antiderivative defined by a definite integral? One of the consequences of the Mean Value Theorem is that any function whose derivative is zero on an interval must be constant on the interval. If $g'(x) = 0$ for all $x \in[a,b]$, then
$$so $g(b) = g(a)$.
$$If $F'(x) = f(x)$, then $\int_a^x f(t)\,dt - F(x)$ has derivative zero. Therefore, there is constant $C$ such that
$$If we set $x=a$, then the integral is zero, and we get the value for $C$,
$$This tells is that
Note that this is valid for any antiderivative. This makes the desired connection between integration as antidifferentiation and Cauchy's definition of integration as the limit of approximating summations. For this reason, the original name of this connection between the two ways of thinking about integration is the Fundamental Theorem of Integral Calculus. Rather unfortunately, calculus textbooks in the 1960s shortened the name to the Fundamental Theorem of Calculus, leading to a de-emphasis of this theorem as providing the connection between two very different ways of thinking about integration.
Why they are called Riemann sums
I quote from my book A Radical Approach to Lebesgue's Theory of Integration):
Bernard Riemann received his doctorate in 1851, his Habilitation in 1854. The habilitation confers recognition of the ability to create a substantial contribution to research beyond the doctoral thesis, and it is a necessary prerequisite for appointment as a professor in a German university. Riemann chose as his habilitation thesis the problem of Fourier series. It was titled Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe (On the representability of a function by a trigonometric series), and, strictly speaking, it answered the broader question: When can a function over (-π,π) be represented as a series of the form
a_0/2 + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin (nx) \right)?
This is where we find the Riemann integral, introduced in a short section before the main body of the thesis, part of the groundwork that he needed to lay before he could tackle the real problem of representability by a trigonometric series.
\vdots
Riemann devotes three brief pages to the definition of the definite integral, the definition of an improper integral, and the statement and proof of the necessary and sufficient condition for integrability. He then spends one page describing a function that is discontinuous at every rational number with an even denominator but which is integrable, thus showing that while continuity is a sufficient condition for integrability, it is far from necessary.
Riemann's precise definition of the approximating sum for a function f defined on [a,b] requires choosing a partition of the interval,
and a set of tags $x_1^* \in [x_0,x_1]$, $x_2^* \in [x_1,x_2]$, \ldots, $x_{n}^* \in [x_{n-1},x_n]$. The Riemann sum is then given by
$$This definition has confused students for decades. It was never intended to be the basis for a working definition of the integral. Its purpose was to enable Riemann to discover the following necessary and sufficient conditions for a function to be integrable. It is admirably suited to this purpose.
Riemann’s necessary and sufficient conditions for integrability. Let f be a bounded
function on [a, b]. This function is integrable over [a, b] if and only if for any σ > 0, a
bound on the oscillation, and for any ν > 0, a bound on the sum of the lengths of the
intervals where the oscillation exceeds σ, we can find a δ response so that for any partition of [a, b] with subintervals of length less than δ, the subintervals on which the oscillation is
≥ σ have a combined length that is < ν.
This answered Riemann’s question and enabled him to proceed with his investigation of
Fourier series. I sincerely hope that no one expects students in their first year of studying
calculus to absorb this result.
David Bressoud is DeWitt Wallace Professor Emeritus at Macalester College and former Director of the Conference Board of the Mathematical Sciences. Information about him and his publications can be found at davidbressoud.org