Riemann Integrals proof I'm working on a question involving Riemann Sums, and I'm not quite sure how to take the information I've gotten and prove an integration. The question prompts me this: Let $f(x) = 2x +1$ on $[0,1]$ Let p be the partition $0 < 1 / n < 2 / n < · · · < n / n$. Compute limn→∞ S¯(p) and limn→∞ S_(p) For this I got these answers: S¯(p) = $(2j / n) + 1$ and S_(p) = $(2(j-1) / n) + 1 $. I am then asked to use this information to prove that $\int_{0}^{1} 2x + 1 dx = 2$. However I am not exactly sure how to use this information to prove this. Most of the proofs I've done so far has been proving whether a function is integrable or not, not to actually prove the area of an integral. Any help is greatly appreciated!

Let

* $\displaystyle\mathscr{P} = \left\\{ 0,\frac{1}{n},\frac{2}{n},\cdots,\frac{n}{n}\right\\}$ be a partiton of $[0,1]$.

* $M(f;I_{k})= \sup \left\\{\: f(x) \ : \ x \in I_{k}\ \right\\}$

* $|I_{k}| =\text{Length of the interval $I_{k}$}$




The Upper Riemann Sum corresponding to $\mathscr{P}$ is defined as \begin{align*} U\left(f;\mathscr{P}\right) &= \sum_{k=1}^{n} M(f;I_{k})\cdot |I_{k}| \\\&= \left( \frac{2}{n}+1\right) \cdot \frac{1}{n} + \left(\frac{4}{n}+1\right)\cdot \frac{1}{n} + \cdots + \left(\frac{2n}{n}+1 \right)\cdot \frac{1}{n} \\\ &= \frac{2}{n^{2}} \cdot (1+2+ \cdots +n) +1 \\\ &= \frac{2}{n^{2}} \cdot \frac{n^{2}+n}{2} + 1 \end{align*}

Now take $\displaystyle\lim_{n \to\infty} U\left(f,\mathscr{P}\right)$ to get the answer.