__TRON__
01-10-2007, 12:17 PM
howdy, I'm a real novice with a simple question relating to method of exhaustion and area of parabolas (in this example using outer rectangles).
Given a parabola with base b divided into n sections, where x = kb/n, and the curve is found at (kb/n)², the area under the curve is described by
L x W = (b/n) • (kb/n)² = (b³/n³)k. Clear so far.
And so to approximate the sum of all these sections, would be described by:
Sn = (b³/n³) • (1² + 2² +...+ n²)
Again, clear so far.
Okay, so the text from which I'm reading is called "Calculus, Volume I" by a Tom M. Apostol, and at this point he says "there is an interesting identity which makes it poss. to evaluate this sum [the squares of all values for k] in a simpler way, namely
1² + 2² +...+ n² = n³/3 + n²/2 + n/6"
And he goes on to talk about a proof of this, and says it's valid for every value n such that n >= 1.
What confuses me is this following text:
"Start with the formula (k + 1)³ = k³ + 3k² + 3k + 1 and rewrite it in the form
3k² + 3k + 1 = (k + 1)³ - k³"
What confuses me is not the formula itself, it's a common enough product, but what confuses me is WHY this formula was selected. My background in math is weak enough that I'm not familiar with most of the formulas related to parabolas and such, does (k + 1)³ have some special significance? Thanks in advance.
Given a parabola with base b divided into n sections, where x = kb/n, and the curve is found at (kb/n)², the area under the curve is described by
L x W = (b/n) • (kb/n)² = (b³/n³)k. Clear so far.
And so to approximate the sum of all these sections, would be described by:
Sn = (b³/n³) • (1² + 2² +...+ n²)
Again, clear so far.
Okay, so the text from which I'm reading is called "Calculus, Volume I" by a Tom M. Apostol, and at this point he says "there is an interesting identity which makes it poss. to evaluate this sum [the squares of all values for k] in a simpler way, namely
1² + 2² +...+ n² = n³/3 + n²/2 + n/6"
And he goes on to talk about a proof of this, and says it's valid for every value n such that n >= 1.
What confuses me is this following text:
"Start with the formula (k + 1)³ = k³ + 3k² + 3k + 1 and rewrite it in the form
3k² + 3k + 1 = (k + 1)³ - k³"
What confuses me is not the formula itself, it's a common enough product, but what confuses me is WHY this formula was selected. My background in math is weak enough that I'm not familiar with most of the formulas related to parabolas and such, does (k + 1)³ have some special significance? Thanks in advance.