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I know I definitely saw an example of this in this site in the past, but I can no longer find it.

In many (dare I say most?) Calc. I classes, if I, say, wanted to evaluate $$\int_{a}^{b}f(x)\text{ d}x$$ I would split the interval $[a, b]$ into $n$ equally-spaced rectangles, each with height $f(x_i)$ ($i \geq 1$), where $x_i = a + b \cdot \Delta x$ for $i \geq 1$, $\Delta x = \dfrac{b-a}{n}$, $x_0 = a$, $x_n = b$, and compute $$\lim_{n \to \infty}\sum_{i=0}^{n}f(x_i)\,\Delta x\text{.}$$ I seem to recall seeing a problem on this website where $f$ isn't a nice polynomial function, and it was beneficial to use a different partition from the equally-spaced $n$ rectangles. I unfortunately cannot find this now, after searching.

Stewart's text does not appear to have examples of other partitions for computing Riemann sums. Where can I find examples of definite integrals computed using other partitions besides the $n$ equally-spaced rectangles partition?

quid
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Clarinetist
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2 Answers2

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For evaluating the integral of polynomials (or linear combination of powers of $x$) it is best to have a non standard partition and the points may be chosen to be in geometric progression. Thus $x_{i} =ar^{i} $ and $r^{n} =b/a$ and $r\to 1$ as $n\to\infty$. This works if $a, b$ are of same sign and can be extended to general $a, b$ with a little more effort. If you try this approach with $f(x) =x^{m} $ then you immediately get the integral as $(b^{m+1}-a^{m+1})/(m+1)$ for $m\neq - 1$ (see details in this answer). When you use the same technique with $f(x) = x^{-1}$ you get the limit $$\lim_{n\to\infty} n(\sqrt[n] {b/a} - 1)$$ and this is nothing but $\log (b/a)$ (see details in this answer).

Paramanand Singh
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  • I think Fermat did this. – marty cohen Sep 17 '17 at 19:17
  • @martycohen: I don't know. If you have any interesting reference relating to its history you may add here. But this particular approach is not well advertised in textbooks. The limit of a Riemann sum almost always involves points of partition in arithmetic progression. – Paramanand Singh Sep 17 '17 at 19:22
  • Look in any biography of Fermat. See my answer for a reference. – marty cohen Sep 17 '17 at 19:30
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Here is Fermat's method for integrating powers:

https://mathcs.clarku.edu/~ma121/fermat.pdf

marty cohen
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