I just started learning this topic and I'm quite confused about the methods that are being used to create the riemann sums. The method I tried using involves calculating the integral as the sum:
$$\lim_{n\to \infty}\sum_{i=1}^n f(x_i^*)\Delta x$$
where $x_i^*=a+i\Delta x$ and $\Delta x=\frac{b-a}{n}$. Using this I tried evaluating the integral
$$\int_1^2 {\frac{1}{x}}dx$$
$$\Delta x=\frac{2-1}{n}=\frac{1}{n},\hspace{0.3cm} x_i^*=a+i\Delta x=1+\frac{i}{n}=\frac{i+n}{n},\hspace{0.3cm} f(x_i^*)=\frac{n}{n+i}$$
All I need to do is calculate $$\lim_{n\to\infty}\sum_{i=1}^n \frac{1}{n+i}$$
Instead in couple of other books using the same exact example they provide completely different solution.
„Subdivide the interval [1,2] into n parts so that the points of division $x_i$ $(i=0, 1, 2, \ldots, n)$ form the geometric progression:
$$x_0=1,\hspace{0.2cm} x_1=q,\hspace{0.2cm} x_2=q^2,\hspace{0.2cm} x_3=q^3 \hspace{0.2cm}\ldots\hspace{0.2cm} x_n=q^n=2, $$ whence $q=\sqrt[n]2$ “
There is another way to divide this interval. It involves using $x_i=2^{\frac{i}{n}}$.
I just don't understand what's the motivation behind these solutions. Frankly I'm not sure how you calculate $x_i^*$ and $\Delta x$ after doing these interval divisions.
Also using the geometric progression divsion seems to be quite popular in some similar problems.