
Calculus Calculus Math Forum 
 LinkBack  Thread Tools  Display Modes 
August 29th, 2017, 12:35 PM  #1 
Member Joined: Feb 2017 From: East U.S. Posts: 40 Thanks: 0  Use the definition of the integral as a limit of a Riemann sum
Equation: "integral from 0 to 6 of (x^2)+36" I know how to find it the easy way... Ex.) I know to take the integral (x^3)/3+36x and evaluate it from 0 to 6 which = (216/3) + 216 == [144] So I know the answer, I just don't understand what my teacher wants. I know there's an "n(n+1)(2n+1) that's supposed to be thrown in there. ANY help would be greatly appreciated! 
August 29th, 2017, 01:04 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,835 Thanks: 733 
What the teacher wants if for you to divide the interval (0,6) into a lot of evenly spaced intervals and set up the Riemann sum over these intervals. Evaluate the sum and then let the number of intervals become infinite and see that the sum converges to the integral.

August 29th, 2017, 01:13 PM  #3  
Member Joined: Feb 2017 From: East U.S. Posts: 40 Thanks: 0  Quote:
 
August 29th, 2017, 02:53 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 21,028 Thanks: 2259 
Using 6$n$ intervals, each of width 1/$n$, the integral can be approximated as $\displaystyle \sum_{k=1}^{6n} ((k/n)^2 + 36)(1/n) = ((6n)(6n + 1)(12n + 1)/(6n^2) + 36(6n))/n$, which tends to 6(12) + 216 = 144 as $n \to \infty$. 
September 3rd, 2017, 11:01 AM  #5 
Member Joined: Feb 2017 From: East U.S. Posts: 40 Thanks: 0  Thanks for your help. I have no idea what you did, but I guess I'll learn soon! 

Tags 
definition, integral, limit, riemann, sum 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Riemann sum limit.  hyperbola  Calculus  2  September 28th, 2015 01:14 PM 
Riemann Integral from definition  azeret  Calculus  3  January 19th, 2011 11:43 PM 
The Riemann zeta function, wrong definition?  nizzeberra  Number Theory  2  November 19th, 2009 05:37 AM 
Definite integral by the limit definition  a.sundar23  Calculus  1  February 23rd, 2009 07:22 PM 
Riemann vs. RiemannStieltjes integral  cheloniophile  Real Analysis  1  November 23rd, 2008 05:30 PM 