
Calculus Calculus Math Forum 
 LinkBack  Thread Tools  Display Modes 
December 23rd, 2010, 11:01 AM  #1 
Member Joined: Dec 2010 Posts: 51 Thanks: 0  Area under a parabolic section (justification)
Hello. I need help with the justification of (n^3)/3 in a particular pair of inequality. In other words, why (n^3)/3 is chosen instead of something else. Also, if possible, I need the proof deduction, not induction (I've already seen it). But first the premise to the question. The sum that underapproximates the area under a parabolic section: The sum that overapproximates the area under a parabolic section: We're only interested in the series in each equation. That is: (p and v are arbitrarily picked) Using Let K = 1, 2, ..., (n1) and manipulating the equation yield:  p < m < v, since p underapproximates the area [if multiplied by ], v overapproximates the area [if multiplied by ], and m is the area [if multiplied by ] that is between the two approximations. The pair of inequalities are: Now, here's my questions) why is m set to ? Why is chosen as a consequence of p and v? Is it because of the end behavior of p and v? That is, as we partition more and more of the horizontal base (n approaches infinity) we approach ? Thanks in advance! The book I'm reading just shows the inequalities without justification for it. 
December 24th, 2010, 11:57 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,095 Thanks: 1905 
What matters is that as n tends to infinity and approach the same limit, which is the area (since is an underestimate and is an overestimate). The common limit can be worked out without introducing a new variable m at all, but the author happened to prefer to use m. Note that implies 

Tags 
area, justification, parabolic, section 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
What's the justification of the imaginary axis' structure?  Tau  Complex Analysis  6  February 13th, 2014 08:53 AM 
area of the green section  Albert.Teng  Algebra  3  October 30th, 2012 09:04 PM 
parabolic problems  suomik1988  Algebra  3  December 1st, 2009 05:24 PM 
parabolic PDE  uri  Applied Math  1  October 11th, 2008 01:21 PM 
Hyperbolic or Parabolic  Blanshan91  Algebra  2  July 16th, 2008 11:22 AM 