I'm learning about integrals and in particular upper and lower Riemann sums. Now in an example I came about the following:
$$L(f, P_n) = \sum_{i = 1}^{n}(x_{i - 1})^2\Delta x = \frac{a^3}{n^3}\sum_{i=1}^{n}(i - 1)^2 =\ ...$$
I don't understand how this equality comes about, in particular $\frac{a^3}{n^3}$ seem to appear out of thin air at the moment.
Any hints, definitions or theorems I shoud consult to get a feeling for this?
As you are integrating $x^2$ with respect to $x$ from $x=0$ to $x=a$, you divide the area into $n$ strips of width $\Delta x=\frac an$ unit each. Your $x_{i-1}$ is consequently $x_{i-1}=(i-1)\Delta x =\frac{a}{n}(i-1)$, so you have $(x_{i-1})^2=\left(\frac{a}{n}(i-1)\right)^2$. Simplifying, you have $$(x_{i - 1})^2\Delta x = \frac{a^3}{n^3}(i - 1)^2 $$