# Thread: Help Summing Difficult Function

1. ## Help Summing Difficult Function

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 solve without having 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?

2. Originally Posted by wolfman29
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
$\displaystyle \Delta x=\frac{r}{n}$
$\displaystyle \lim_{n\to\infty}\frac{r}{n}\sum_{i=0}^n\sqrt{r^2-x_i^2}\Delta x$$\displaystyle =\int_0^r {\sqrt {r^2 - x^2 } dx} = \frac{{\pi r^2 }}{4}$
It is the area of one-fourth a circle.

3. Well, yes, I know that. But what I am trying to do is trying to solve it assuming I don't know the Fundamental Theorem of Calculus (which I do, but this is just a curiosity).

Is there a way to solve this without just saying, "Oh, by definition, it's the area of a quarter of a circle, and thus it is (pi*r^2)/4"?

4. Originally Posted by wolfman29
Well, yes, I know that. But what I am trying to do is trying to solve it assuming I don't know the Fundamental Theorem of Calculus (which I do, but this is just a curiosity).

Is there a way to solve this without just saying, "Oh, by definition, it's the area of a quarter of a circle, and thus it is (pi*r^2)/4"?
You are missing the point.
You simply must recognize that as an approximating sum for that integral.

5. I understand that. But 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?