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December 23rd, 2010, 10:01 AM   #1
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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 under-approximates the area under a parabolic section:


The sum that over-approximates 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, ..., (n-1) and manipulating the equation yield:



---
p < m < v, since p under-approximates the area [if multiplied by ], v over-approximates 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.
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December 24th, 2010, 10:57 PM   #2
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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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