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December 23rd, 2010, 10:01 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, 10:57 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,731 Thanks: 1808 
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 

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area, justification, parabolic, section 
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