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We are asked to get to approximate the area between the function $f(x) = x^{2}$ and the $x$-axis from $x = 0$ to $x = 2$ in $10$ regular partitions. All types of Riemann sums are asked: Left endpoint, right endpoint, and the midpoint Riemann sum.

The width can be observed to be $\frac{2 - 0}{10} = \frac{1}{5}$.

For the left endpoint Riemann sum, we can write and simplify as

$$A = \frac{1}{5}\sum_{i = 1}^{10}f\left(\frac{i-1}{5}\right)$$

When evaluated one by one, we get

$$A=\frac{1}{5}\left(\left(\frac{0}{5}\right)^{2} + \left(\frac{1}{5}\right)^{2} + \left(\frac{2}{5}\right)^{2} + \left(\frac{3}{5}\right)^{2} + \cdots + \left(\frac{7}{5}\right)^{2} + \left(\frac{8}{5}\right)^{2} + \left(\frac{9}{5}\right)^{2}\right)$$

Solving for this, we get $\frac{57}{25}$. However, it seems as if the process is too tedious. Is there a much more easier way for problems like this?


Edit: I am looking for ways to simplify the process of solving the Riemann sum of a general Riemann integrable function, not for an easier solution of the function that I mentioned.

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    $\begingroup$ For this particular problem, there is a formula for $\sum k^2$ here. Calculating a Riemann sum is tedious for a general $f$. $\endgroup$ May 17, 2021 at 9:55
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    $\begingroup$ To avoid such ambiguities, I guess I have to include that my question is for a general Riemann integrable function. $\endgroup$
    – soupless
    May 17, 2021 at 9:59
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    $\begingroup$ Asking if there is a general "formula" for the result of a Riemann sum is quite literally equivalent to asking if there is a general "formula" to evaluate an integral. There isn't. However, hopefully your teacher allows use of a calculator. Techniques for simplifications are case-by-case. $\endgroup$
    – Graviton
    May 17, 2021 at 10:01
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    $\begingroup$ @ArcticChar: there is a formula for summing sines (or cosine) of angles in arithmetic progression. $\endgroup$
    – Paramanand Singh
    May 17, 2021 at 10:01
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    $\begingroup$ From a numerical standpoint, this is one of those calculations (like extracting square roots) that quickly become something you turn over to a computer. In the time it takes you to calculate a sum for a measly $n=2$ or $3$, a computer can do $n=10000$, or many orders of magnitude more, with slices so fine that error approximations are essentially moot. ... That said, simple cases like $f(x)=x^n$ are good to work out symbolically; they put those precalculus sum-of-powers formulas to good use, provide more practice in applying limits, and reveal the elegant patterns that bind things together. $\endgroup$
    – Blue
    May 17, 2021 at 10:27

1 Answer 1

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There is no "general" formula to get the result of a Riemann sum. However,

  • For functions of the form $x^{n}$, Faulhaber's formula may be used.
  • For functions $\sin x$ or $\cos x$, the product-to-sum formula may be used to simplify the entire sum. See this page to get the full answer.
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