Finding the Riemann Sum for the function $\sin(x)$ over an interval

Question

I have been asked to determine the Lower Riemann Sum for the function $f(x) = \sin(x)$ in the interval $(0,j]$. I understand the method, but I just wanted to check if the following was the right answer:

$$\sum_{i=1}^n \sin\left(\frac{ji-j}{n}\right)\cdot\frac{j}{n}.$$

Thanks.

Answer 1

This sum $$\sum_{i=1}^n \sin\left(\frac{ji-j}{n}\right)\cdot\frac{j}{n}$$ is not necessarily the lower Riemann Sum because $\sin x$ is not monotone on (0, j].

It is the Left Riemann Sum in which you have selected the left side points $ \frac{ji-j}{n} $ for evaluation.