Abstract
After what has been said in the last lecture, if we divide \( X-x_0 \) into infinitely small elements \( x_1-x_0, x_2-x_1, \dots , X-x_{n-1}, \) the sum
will converge toward a limit represented by the definite integral.
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Notes
- 1.
Cauchy’s 1823 and 1899 editions both read,
$$\begin{aligned} S=(x_1-x_0)f(x_0)+(x_1-x_2)f(x_1)+\cdots +(X-x_{n-1})f(x_{n-1}),\qquad {(1)} \end{aligned}$$a clear misprint which has been corrected here.
- 2.
Cauchy is clearly allowing \(\theta =0\) and \(\theta =1 \) in this instance—describing the closed interval [0, 1]. His wording usually leaves us in doubt as to whether he is describing an open or closed interval, as he generally uses imprecise language throughout the text to convey his intent. Phrases such as “contained between the limits ...,” “less than ...,” “included between the limits ...,” or simply “between the limits ...,” are used in nearly every part of Calcul infinitésimal—including some of his important theorems. As an example of his inexplicit language, in Lecture Four Cauchy writes, “...contained between the limits \( -\pi /2, +\pi /2,\)” to seemingly describe the domain of both the sine function \(\big (\)whose domain is actually \(\big [-\pi /2, +\pi /2\big ]\big )\) and the tangent function \(\big (\)with an actual domain of \((-\pi /2, +\pi /2) \big )\) within the same sentence!
- 3.
Cauchy often leaves very few clues as to his methods for solving given example problems in his lectures. However, here we can use his intermediate results to determine whether he is using equation (7) with an arithmetic progression or equation (9) with a geometric progression by his use of i for the former, or \(\alpha \) for the latter.
- 4.
Cauchy has just proven the Mean Value Theorem for Definite Integrals. This is actually a special case of the General Mean Value Theorem for Definite Integrals which he will also prove in the next lecture.
- 5.
The 1899 reprint has \(\frac{1}{100}.\)
- 6.
Both of Cauchy’s texts have f(x) here.
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Cates, D.M. (2019). FORMULAS FOR THE DETERMINATION OF EXACT OR APPROXIMATE VALUES OF DEFINITE INTEGRALS.. In: Cauchy's Calcul Infinitésimal. Springer, Cham. https://doi.org/10.1007/978-3-030-11036-9_22
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