Riemann Sums

How can I estimate the areas of the left and right side for Riemann Sums??! This is really confusing me.

For example $f(x) = 5/x$

Quote:

Originally Posted by skyslimit

How can I estimate the areas of the left and right side for Riemann Sums??! For example $f(x) = 5/x$

Over what interval?

Quote:

Originally Posted by skyslimit

How can I estimate the areas of the left and right side for Riemann Sums??! This is really confusing me.

For example $f(x) = 5/x$

You have no interval, so I will do it for the interval $[a,b]$ . So we would have that $\Delta{x}=\frac{b-a}{n}$ , where that is n partitions of the region. We would also have that $f\left(a+\Delta{x}i\right)=f\left(a+\frac{b-a}{n}i\right)=\frac{5}{a+\frac{b-a}{n}i}$ . So we would have that $\int_a^{b}\frac{5}{x}dx\approx\sum_{i=1}^{n}\frac{ 5}{a+\frac{b-a}{n}i}\cdot\frac{b-a}{n}$ and that $\int_a^b\frac{5}{x}dx=\lim_{n\to\infty}\sum_{i=1}^ {n}\frac{5}{a+\frac{b-a}{n}i}\cdot\frac{b-a}{n}$ .

To illustrate the point $\int_1^{e}\frac{5}{x}dx=5$ and $\left|\int_1^{e}\frac{5}{x}dx-\sum_{i=1}^{500}\frac{5}{1+\frac{e-1}{500}i}\cdot\frac{e-1}{500}\right|<.006$

Interval 1 to 5, I need to find the left and right a estimated sides..