Exhaustion Methods Explained: Sum of Areas of Outer & Inner Rectangles

[andrewkg]

Well if anyone has read Apostol's vol 1. Can anyone explain in the beginingthe sum of the areas of the outer rectangles is (b^3/n^3)(k^2) or
(b^3/n^3)(1^2+2^2+3^2…+n^2)

And the area of the inner rectangles and (b^3/n^3)(1^2+2^2+3^2…+(n-1)^2)

What I would like to know is why does the value of k for the inner area sum goes to n-1
Thank you

 

[andrewkg]

Oh yeah. My idea was that the interior rectangles cannot have a value directly at b so the value of k must be under the of n so n doesn't cancek out. Although this woukd not occur anyways because the value of n on the bottom is n^3 and the value of k on tome is only squared. Honestly though I don't have a good idea. I tryed it graphically on my own paper I got lost with what I was doing.

 

[HallsofIvy]

You should know that
$$1^2+ 2^2+ 3^2+ \cdot\cdot\cdot+ n^2= \frac{n(n+1)(2n+1)}{6}$$

As to what happens to the "k", I doubt if anyone can answer because you have not said what "k" represents.

 

[andrewkg]

Nvm I got it. Thanks though

 

[CellCoree]

for your question. The exhaustion method, also known as the method of exhaustion, is a mathematical technique used to calculate the area of a shape by approximating it with smaller and smaller shapes. This method was developed by ancient Greek mathematicians and was refined by mathematicians such as Archimedes and Eudoxus.

In the context of your question, we are using the exhaustion method to calculate the area under a curve. Specifically, we are using rectangles to approximate the area under the curve. The idea is to divide the interval [0,b] into n equal subintervals and then calculate the area of each rectangle. The sum of these areas will give us an approximation of the area under the curve.

Now, let's focus on the inner and outer rectangles. The outer rectangles are the ones that touch the curve at their upper edge, while the inner rectangles are the ones that touch the curve at their lower edge. As the number of rectangles increases (as n increases), the width of each rectangle decreases, resulting in a better approximation of the area under the curve.

In the case of the outer rectangles, the value of k represents the height of each rectangle. As n increases, k also increases, but the value of k for the inner rectangles is always one less than the value of k for the outer rectangles. This is because the inner rectangle at the rightmost end of the interval [0,b] does not touch the curve at its upper edge, but rather at its lower edge. Therefore, the value of k for this rectangle is n-1.

To sum up, the value of k for the inner rectangles goes to n-1 because the rightmost inner rectangle does not touch the curve at its upper edge, resulting in a one unit decrease in the value of k. This is a fundamental aspect of the exhaustion method and is crucial in accurately calculating the area under a curve. I hope this explanation helps clarify your question.