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I'm attempting to calculate the right Riemann sum and approximate the area of the region bounded by $f(x) = 25 - x^2$ on the interval $[-5, 5] = [a, b]$.

$$\sum_{k = 1}^{n}{f(a + k\Delta x)}\Delta x$$

I'm taking a table of values for $n = \{10, 30, 60, 80 \}$.

I've attempted to take the closed form of: $$\sum_{k = 1}^{n}{x^2} = \dfrac{n(n+1)(2n+1)}{6}$$ and subtract it from $$\sum_{k = 1}^{n}{25} = 25n$$ such that $$\left(25n - \dfrac{n(n+1)(2n+1)}{6}\right)\frac{n}{10} = \sum_{k = 1}^{n}{f(a + k\Delta x)}\Delta x$$ but I suspect this may be an improper approach.

Results

I get these results, but I'm told that these results are incorrect. I am uncertain of my error.

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  • $\begingroup$ You forgot $\Delta x$ when doing the substitiution back. For example, you don't have a sum $\sum_{k=1}^{n}{25}$, but $\sum_{k=1}^{n}{25\Delta x_k}$. The other summand (the one with $x$ squared) is more problematic than the other one with respect to this new addition. $\endgroup$ – Miel Sharf Apr 28 '15 at 16:28
  • $\begingroup$ @MielSharf So, I'd just take the whole left side and multiply through by $\Delta x$ to have the proper sum? $\endgroup$ – mmam Apr 28 '15 at 16:36
  • $\begingroup$ Well, you need to sum on the $\Delta x$-s. For example, if you look at the summand $\sum_{k=1}^{n}{x^2 \Delta x}$, say with 30 partition intervals, then $\Delta x$ = 1/3 (because your interval is of length 10, and there are 30 even smaller intervals). Furthermore, $x_k = -5+\frac{k}{3}$ (because your interval starts at -5, and $\Delta x$ = 1/3) so you end up with the sum $\sum_{k=1}^{n}{x^2 \Delta x}= \sum_{k=1}^{n}{((-5+\frac{k}{3})^2 \cdot 1/3)}$ $\endgroup$ – Miel Sharf Apr 28 '15 at 16:52
  • $\begingroup$ @MielSharf look at what I've put in my answer, if you would $\endgroup$ – mmam Apr 28 '15 at 16:58
  • $\begingroup$ I see , you forgot to shift your interval - the equality $\sum_{k=1}^{n}{k^2}=\frac{n(n+1)(2n+1)}{6}$ refers to summing on the natural numbers. You need to sum on your partition points, which are $x_k = -5+\frac{points}{n}\cdot k$ $\endgroup$ – Miel Sharf Apr 28 '15 at 17:00

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