Hey everyone, this is my first thread, and I hope it will give me a good experience on this site.

Anyway, onto the question:

I have been trying to figure out a way to sum this:

$\displaystyle \lim_{n\to\infty}\frac{r}{n}\sum_{i=0}^n\sqrt{r^2-x_i^2}\Delta x$

Where

$\displaystyle x_i=\frac{ri}{n}$

and

$\displaystyle \Delta x=\frac{r}{n}$

The next step is

$\displaystyle \lim_{n\to\infty}\sum_{i=0}^n(\sqrt{r^2-\frac{r^2i^2}{n^2}})(\frac{r}{n})$

I did some manipulating, and got:

$\displaystyle \lim_{n\to\infty}\frac{r}{n}\sum_{i=0}^n\sqrt{r^2-\frac{r^2i^2}{n^2}}$

All the help I have gotten has told me that the answer is zero, but that can't be possible, because I started with the area under the curve of the basic unit circle in the first quadrant,

$\displaystyle y=\sqrt{r^2-x^2}$

and I have not made any mistakes so far.

So how can I come up with a non-zero answer for this?

EDIT:I want to solvewithouthaving to use the fundamental theorem of calculus, and just saying that it is equal to $\displaystyle \frac{\pi r^2}{4}$

EDIT 2:Just like for

$\displaystyle \sum_{i=0}^n i=\frac{n(n+1)}{2}$,

is there a way I can do that sort of method the problem I have presented?