How can I estimate the areas of the left and right side for Riemann Sums??! This is really confusing me.
For example $\displaystyle f(x) = 5/x $
You have no interval, so I will do it for the interval $\displaystyle [a,b]$. So we would have that $\displaystyle \Delta{x}=\frac{b-a}{n}$, where that is n partitions of the region. We would also have that $\displaystyle f\left(a+\Delta{x}i\right)=f\left(a+\frac{b-a}{n}i\right)=\frac{5}{a+\frac{b-a}{n}i}$. So we would have that $\displaystyle \int_a^{b}\frac{5}{x}dx\approx\sum_{i=1}^{n}\frac{ 5}{a+\frac{b-a}{n}i}\cdot\frac{b-a}{n}$ and that $\displaystyle \int_a^b\frac{5}{x}dx=\lim_{n\to\infty}\sum_{i=1}^ {n}\frac{5}{a+\frac{b-a}{n}i}\cdot\frac{b-a}{n}$.
To illustrate the point $\displaystyle \int_1^{e}\frac{5}{x}dx=5$ and $\displaystyle \left|\int_1^{e}\frac{5}{x}dx-\sum_{i=1}^{500}\frac{5}{1+\frac{e-1}{500}i}\cdot\frac{e-1}{500}\right|<.006$