Help with converting integral to Riemann Sum

Question

I have the integral $$\int_1^3\sin(x)\,\mathrm dx$$ and I want to convert it to a Riemann Sum. I understand that the first thing I should do is set the limit as $n\to\infty$. I am stumped as what I should do next. Can someone walk me through the steps of converting integrals to Riemann Sums?

Answer 1

You have to introduce the subdivision points. As the intercal is $[1,3]$, with length $2$, we obtain: $$x_0=1,\:x_1=1+\frac2n,\dots,\: x_i=1+i\cdot \frac 2n,\dots,\: x_n=1+n\cdot \frac 2n=3$$

Then, the Riemann sums associated with this subdivision are \begin{align} s_n&=\sum_{i=0}^{n-1}\sin x_i\,\cdot\frac2n=\frac2n\sum_{i=0}^{n-1}\sin\Bigl(1+\frac{2i}n\Bigr),\\[1ex] S_n&=\frac2n\sum_{i=1}^{n}\sin\Bigl(1+\frac{2i}n\Bigr). \end{align}