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The integral

For the whole of the 18th century and into the 19th, integration had been treated as the inverse of differentiation. Cauchy's definition of the derivative given above makes the following clear:

• the derivative will not exist at a point for which the function is discontinuous
• yet the integral may afford no difficulty, since even discontinuous curves may determine a well-defined area.

The fact that the inverse could not always be computed exactly led 18th mathematicians to do much work approximating the values of definite integrals:

Euler treated sums of the form $\sum_{k = 0}^n f(x_k) (x_{k+1} - x_k)$

as approximations to the integral $\int_{x_0}^{x_n} f(x) dx$

Poisson attempted a proof of the following what he called the fundamental proposition of the theory of definite integrals, which he stated as follows:

If the integral F is defined as the antiderivative of $f$, and if $b - a = nh$

can it be proved that F(b) - F(a) is the limit of the sum

$S = hf(a) + hf(a + h) + . . . + hf(a + (n - 1)h)$ ::as $h$ gets small. In effect, Poisson was the first to attempt a proof of the equivalence of the antiderivative and limit-of-sums conceptions of the integral. Following this tradition, Cauchy also defined the definite integral in terms of the limit of the integral sums. Then, having defined the integral independently of differentiation, it was necessary for him to prove the usual relation between the integral and the antiderivative.'"`UNIQ--ref-00000001-QINU`"' He did this as follows:'"`UNIQ--ref-00000002-QINU`"' * Defining the integral as the limit of Euler-style sums $\sum f(x_k)(x_{k+1} - x_k)$ for sufficiently small $x_{k + 1} - x_k$ * Assuming explicitly that it was continuous on the given interval (and implicitly that it was uniformly continuous) * Showing that all sums of that form approach a fixed value, called by definition the integral of the function on that interval * Borrowing from Lagrange the mean-value theorem for integrals, proving the [[Fundamental theorem of calculus]]. Similar views were developed at about the same time by Bolzano.'"`UNIQ--ref-00000003-QINU`"' As mentioned above, the existence of continuous nowhere differentiable functions, including of course the "pathological" functions of Bolzano and Weierstrass, contributed to the concerns about the foundations of analysis. Riemann exhibited a function $f(x)$ with the following characteristics: :it is discontinuous at infinitely many points in an interval and yet its integral exists and defines a continuous function $F(x)$ that, for the infinity of points in question, fails to have a derivative

Cauchy's definition of the integral was guided largely by geometrical feeling for the area under a curve. Riemann's function made clear that the integral required a more careful definition than that of Cauchy. The present-day definition of the definite integral over an interval in terms of upper and lower sums generally is known as the Riemann integral, in honor of the man who gave necessary and sufficient conditions that a bounded function be integrable.

Primary sources

• Cauchy, A.-L. Résumé des leçons données à l’école royale polytechnique sur le calcul infinitésimal, Paris, 1823.
• Lagrange, Traite de la resolution des squations numeriques de tous les degres, 2nd ed., 1808, in [B24, vol. VIII, pp. 46-7, 163].