confusion in writing down my understanding mathematically. Riemann integral

Question

I am going to compute $\int_{-1}^1 f $ for $f(x)= \begin{cases} 0 & -1\leq x\leq 0 \\ x & 0< x\leq 1 \end{cases} $ using the definition of Riemann Integral. Since my textbook does not have a solved example for this topic, I would appreciate it if you take a look at my answer to confirm me if I have understood the contents correctly.

since $f(x)$ is bounded and continuous on $[-1,1]$, then by theorem, it is guaranteed that $f(x)$ is Riemann Integrable on $[-1,1]$.
so it is enough to find any of $ \overline{\int_{a}^b} f$ or $ \underline{\int_{a}^b} f$.
let s compute $ \overline{\int_{a}^b} f =$ inf $ U[f,\sigma]$ where $\sigma$ is a subdivision of $[-1,1]$.
Now how should I choose $\sigma$? should I specify a set like $\{-1,\frac{-1}{2}, 0, \frac{1}{2}, 1\}$? or a general general $\sigma$ with componentinterval length of $\frac{2}{n}$?
I understand the definition, and I remember these integrals from calculus. I do not know for real analysis course ( where I am dealing with sup and inf of f over any component interval ), how am I supposed to mathematically write the answer to these questions. Thank you

Answer 1

Consider the subdivision $\sigma$ of $[-1,1]$ into $2n$ sub-intervals of equal length $1/n$ with end points $-1,-\frac{n-1}{n},\dots, -\frac{k}{n},\dots, -\frac{2}{n},-\frac{1}{n},0,\frac{1}{n},\frac{2}{n},\dots,\frac{k}{n},\dots,1.$

Like you said, $f\in R([-1,1])$ and since $f=0$ in $[-1,0]$,

$U(f,\sigma)=\sum_{i=1}^{2n}M_i\Delta x_i=\frac{1}{n^2}\big(1+2+\cdots+n\big)=\frac{1}{2}\Big(1+\frac{1}{n}\Big)$

so that $\inf U(f,\sigma)=\frac{1}{2}.$