Newton and Leibniz invented integral calculus in the seventeenth century, but for a long time nobody worried about defining precisely what an integral is. For ordinary functions, the integral was simply the inverse of the derivative, or the area under the curve, and that was enough.

Strange functions

In the early nineteenth century the situation changed. Fourier showed that very irregular functions, with jumps and corners, could be written as sums of sines and cosines, and the coefficients of those sums were given by integrals. But what sense did it make to integrate a function full of jumps? Cauchy gave a first definition for continuous functions. Dirichlet, Riemann’s teacher, found in 1829 a function that could not be integrated at all: the one that equals 1 at the rationals and 0 at the irrationals.

Riemann tackled the problem in his habilitation work of 1854, on trigonometric series. To know which functions could be written as Fourier series, he first needed to know exactly which functions could be integrated.

The definition

Riemann’s idea is simple. To integrate $f$ over the interval $[a, b]$:

  1. cut the interval into pieces $[x_0, x_1], [x_1, x_2], \ldots, [x_{n-1}, x_n]$, with $x_0 = a$ and $x_n = b$;
  2. in each piece, choose any point $\xi_i$;
  3. form the sum of the areas of the rectangles with base $x_i - x_{i-1}$ and height $f(\xi_i)$:
$$S = \sum_{i=1}^{n} f(\xi_i)\,(x_i - x_{i-1}).$$

If, as the pieces all become very narrow, these sums approach a single number, no matter how the pieces and the points were chosen, that number is the integral:

$$\int_a^b f(x)\,dx = \lim S.$$

The requirement that the choice of points should not matter is the heart of the definition. In the figure, change the point chosen in each strip and the number of strips:

Which functions are integrable

Riemann also gave a criterion. In each strip, the function varies between a minimum and a maximum value; the difference is the oscillation of the function in that strip. The function is integrable when the sum of the oscillations multiplied by the widths of the strips can be made as small as one wishes. Every continuous function passes the test, and so do many functions with jumps: it is enough that there are not “too many” jumps.

Dirichlet’s function fails: in any strip, however small, there are rationals and irrationals, and the oscillation is always 1. The sums are 1 or 0 depending on the points chosen, and there is no single limit.

Exercise: the left sum for x²

Compute the Riemann sum of $f(x) = x^2$ on $[0, 1]$ with $n$ equal strips and the left point of each strip. Show that it tends to $\frac{1}{3}$.

The strips have width $\frac{1}{n}$, and the left point of strip $i$ is $\frac{i-1}{n}$. The sum is

$$S_n = \sum_{i=1}^{n}\left(\frac{i-1}{n}\right)^2\frac{1}{n} = \frac{1}{n^3}\sum_{k=0}^{n-1}k^2 = \frac{(n-1)n(2n-1)}{6n^3}.$$

As $n$ grows, this approaches $\frac{2n^3}{6n^3} = \frac{1}{3}$. For $n = 10$, $S_{10} = \frac{9 \cdot 10 \cdot 19}{6000} = 0.285$.

Exercise: the step function

Let $f(x) = 0$ for $x < \frac{1}{2}$ and $f(x) = 1$ for $x \ge \frac{1}{2}$. Show that $f$ is integrable on $[0, 1]$ and compute the integral.

The function oscillates only in the strip that contains $\frac{1}{2}$, and there the oscillation is 1. With $n$ equal strips, the sum of oscillations times widths is at most $2 \cdot \frac{1}{n}$ (if $\frac{1}{2}$ falls on the boundary between two strips), which tends to zero. So $f$ is integrable. All the sums approach the area of the rectangle with base $\frac{1}{2}$ and height 1: the integral is $\frac{1}{2}$.

Sources: Bernhard Riemann, “Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe” (habilitation of 1854, published 1867) · Thomas Hawkins, Lebesgue’s Theory of Integration: Its Origins and Development (1970) · Detlef Laugwitz, Bernhard Riemann 1826–1866 (1999).

All topics on Bernhard Riemann · ← The life of Bernhard Riemann · The rearrangement theorem →

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