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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.

soupless
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    For this particular problem, there is a formula for $\sum k^2$ here. Calculating a Riemann sum is tedious for a general $f$. – Arctic Char May 17 '21 at 09:55
  • This is the expected procedure. There is some simplification by using formula for sum of squares of consecutive natural numbers. Also this is not tedious at all and in fact you are not required to write so many terms explicitly. Just use $\sum$ notation. – Paramanand Singh May 17 '21 at 09:56
  • @ArcticChar Sorry, I am referring to a general Riemann integrable function, if there is a 'technique' to easily simplify things. – soupless May 17 '21 at 09:57
  • Try the Riemann sum $\int_0^1 \sin x dx$, I doubt if you can simplify anything. – Arctic Char May 17 '21 at 09:59
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    To avoid such ambiguities, I guess I have to include that my question is for a general Riemann integrable function. – soupless May 17 '21 at 09:59
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    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. – Graviton May 17 '21 at 10:01
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    @ArcticChar: there is a formula for summing sines (or cosine) of angles in arithmetic progression. – Paramanand Singh May 17 '21 at 10:01
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    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. – Blue May 17 '21 at 10:27

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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.
soupless
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