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