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June 12th, 2015, 03:24 PM  #1 
Member Joined: Sep 2013 Posts: 58 Thanks: 0  Fundamental Theorem of Calculus Question
I am preparing for an exam and was looking at a past exam problem. I was really stuck on this question. Can anyone elaborate where the integrand x^2 came from? As well as the limits of integration of 0 and 1, or what this whole 1/N Riemann's sum means in relation to the righthand rule in general. I have no idea where to start. Thanks, Tzad Last edited by skipjack; June 12th, 2015 at 04:52 PM. 
June 12th, 2015, 04:20 PM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
In the Riemann sum definition of the integral we start by dividing the interval of integration (here from 0 to 1) into N subintervals. One can show that the subintervals don't have to be all the same length, but that is can be done and is simplest. Taking the subintervals to be the same length, each will have length 1/N. We then choose a single point, $\displaystyle x_i$, in each interval and find the area of the rectangle with base 1/N and height $\displaystyle f(x_i)$. If we choose that point to always be the right endpoint of the interval, it will be $\displaystyle x_i= \frac{i}{N}$ so the height of that rectangle will be $\displaystyle \left(\frac{I}{N}\right)$ which, here, is $\displaystyle \frac{i^2}{N^2}$. The area of a rectangle with base $\displaystyle \frac{1}{N}$ and height $\displaystyle \frac{i^2}{N^2}$ is $\displaystyle \frac{1}{N}\frac{i^2}{N^2}$. The total area of all those rectangles is the sum $\displaystyle \frac{1}{N}\sum_{i=1}^N \frac{i^2}{N^2}$. That sum could be written as $\displaystyle \frac{1}{N^3}\sum_{i= 0}^N i^2$ but I assume they did not do that in order to make it clear that the "$\displaystyle \frac{i^2}{N^2}$" term is the "$\displaystyle x^2$" term. By the way, You could also do that problem "the other way around", using the fact that $\displaystyle \sum_{i=1}^n i^2= n(2n1)(n 2)$. Then the sum $\displaystyle \frac{1}{N^3}\sum_{i=1}^N i^2= \frac{1}{N^3}N(2N 1)(n N)= \frac{2N^3+ 6N^2+ N}{6N^3}= \frac{1}{3}+ \frac{1}{n^2}+ \frac{6}{N^2}$ which goes to $\displaystyle \frac{1}{3}$ as N goes to infinity. Last edited by Country Boy; June 12th, 2015 at 04:26 PM. 
June 12th, 2015, 04:29 PM  #3 
Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 
Do you know the definition of a Reimann Sum? Let $P = \{x_0,\,x_1,\,x_2,\,...,\,x_n\}$ (where $x_0 < x_1<x_2<...<x_n$) be a partition of $f(x)$ into $n$ subintervals. For each subinterval $[x_{i  1},\,x_i]$ choose a number $c_i$ in this interval. The Reimann sum is defined as $$R = \sum_{i = 1}^{n}(x_i  x_{i  1})f(c_i)$$ Also, if the largest subinterval $[x_{i  1},\,x_i]$ approaches zero length (it does in your case) then $$\lim_{n\to\infty}\,R = \int_{x_0}^{x_n}f(x)\,dx$$ In your specific case, you have $f(x) = x^2$, $c_i = x_i = \dfrac{i}{N}$ and $x_{i  1}  x_i = \dfrac{1}{N}$. Thus your Reimann sum is $$R = \sum_{i = 1}^N\dfrac{1}{N}\left(\dfrac{i}{N}\right)^2 \\ = \dfrac{1}{N}\sum_{i = 1}^N\dfrac{i^2}{N^2}$$ Now, it is easy to see that $x_N = \dfrac{N}{N} = 1$ and $x_0 = x_1  \dfrac{1}{N} = \dfrac{1}{N}  \dfrac{1}{N} = 0$. Thus, $$\lim_{n\to\infty}\,R = \int_0^1x^2\,dx$$ Sorry if the explanation is too rigorous. 
June 12th, 2015, 05:07 PM  #4  
Member Joined: Sep 2013 Posts: 58 Thanks: 0  Quote:
Thanks for the response, I have a couple of questions. 1. I understand that I'm breaking up the function so that each base is 1/N, I also understand that the height is f(xi). When you say the right point will be xi=i/N, are you saying that's its area? Also, I kind of lost you after you said the height of the rectangle is I/N. I don't see how you got that. 2. I also know in an ordinary Reimman's sum, the 1/N (Or Delta X) is on the right side of the f(xi). Can we rearrange this equation so that it is like that? Or is that wrong? 3. I have no idea what you mean about looking at the problem from "the other way around". It seems really cool and I'd like to know more if you can elobarate on how ∑i^2=n(2n−1)(n−2) Thanks, Tzad  
June 12th, 2015, 05:11 PM  #5  
Member Joined: Sep 2013 Posts: 58 Thanks: 0  Quote:
Thanks, Tzad Last edited by Tzad; June 12th, 2015 at 05:18 PM.  
June 12th, 2015, 06:14 PM  #6  
Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244  Quote:
Quote:
$$\lim_{n\to\infty}\,R = \int_{x_0}^{x_n}f(x)\,dx$$ In your case, $x_0 = 0$, $x_n = 1$ and $f(x) = x^2$.  

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