The Riemann sum/integral, is defined to be $$ \int_a^b f(x)dx := \lim_{n,\Delta x_i \rightarrow 0} \sum_{i=1}^n f(x_i^*)\Delta x_i $$ whenever the sum exists, where $\Delta x_i$ is the sub-interval width for the $i$'th segment and $x_i^*$ is an arbitrary number in the sub-interval $i$. Let's quickly review a few examples before discussing the problems.
Example 1: $\lim_n \sum_i (\frac{1}{n})^{\frac{3}{2}} \sqrt{n+3i} = \lim_n \sum_i \frac{1}{n}\frac{1}{\sqrt{n}} \sqrt{n+3i} = \lim_n \frac{1}{n} \sqrt{1+\frac{3i}{n} } = \int_0^1 \sqrt{1+3x} dx =\frac{14}{9}. $
Example 2: $\lim_n \sum_i \frac{n}{n^2+i^2} = \lim_n \sum_i \frac{1}{n} \frac{1}{1+ (\frac{i}{n})^2} = \int_0^1 \frac{1}{1+x^2} = \frac{\pi}{4}. $
Example 3: $\lim_n (\frac{1}{1+\sqrt{n}} + \frac{1}{2+\sqrt{2n}} + \dots +\frac{1}{n+\sqrt{n^2}} ) = \lim_n \sum_i \frac{1}{i+\sqrt{in}} = \lim_n \sum_i\frac{1}{n} \frac{1}{ \frac{i}{n} + \sqrt{\frac{i}{n} } } = \int_0^1 \frac{dx}{x + \sqrt{x}} = 2 \ln 2.$
The common pattern in all these examples reveals a couple of restrictions: 1) the summation variable has to be in some form of $(\frac{i}{n})$ which, 2) necessarily forces the integral limits $\int_0^1$. The general Riemann sum stated at the beginning of the post doesn't suffer from these restrictions. So,
Question: What are some of the insightful examples of Riemann sums where the integration interval is other than $[0,1]$ and the summation variable is not necessarily $(\frac{i}{n})$? How do we replace such a summation variable with $x$ and $dx$ and calculate the sums?
The following sums may be such examples, which I couldn't recast in the form described above. I'd appreciate any help with these sums as well:
Example 4: $\lim_n \frac { [\frac{5}{2}] + [(\frac{5}{2})^2 ] + \cdots + [ (\frac{5}{2})^n] } {(\frac{5}{2})^n}$ where $[\cdot]$ is $floor(\cdot)$,
Example 5: $\lim_n (\frac{1}{\sqrt{n^2 +n}} + \frac{1}{\sqrt{n^2 +n+1}} + \cdots + \frac{1}{\sqrt{n^2 +2n}} )$
