# Apostol's Archimedes area proof for a parabola

• gom2
In summary, Apostol's proof in his book Calculus Vol I 2nd ed has a contradiction which can only be resolved by someone who has the book.
gom2

## Homework Statement

My question is on the last bit of Apostol's proof, in his book Calculus Vol I 2nd ed, where he shows that the area under the parabola = b$$^{3}$$/3 where b is the base of the rectangle enclosing the parabola.

The bit I am confused about is where his contradiction n$$\geq$$b$$^{3}$$/(A-b$$^{3}$$/n) came from?

I know this is impossible for anyone to solve unless they own the book... However, please please could someone understand my frustration and also the lack of knowledge of knowing Latex enough to replicate the problem. If you own the book, please could you help me?

I would greatly appreciate help!

## The Attempt at a Solution

I have tried various approaches in trying to attain the n$$\geq$$b$$^{3}$$/(A-b$$^{3}$$/n), however, I have not been successful either way. Sorry, it might not seem like proper attempts have been made, but I have tried, and I have constantly failed to gain it.

It's just from the fact that you can pick a number of partitions n that is greater than or equal to $$\frac{b^3}{A-\frac{b^3}{n}}$$ so A < $$\frac{b^3}{3}$$ is false. The same logic works for the case A > $$\frac{b^3}{3}$$, just in reverse

Feldoh said:
It's just from the fact that you can pick a number of partitions n that is greater than or equal to $$\frac{b^3}{A-\frac{b^3}{n}}$$ so A < $$\frac{b^3}{3}$$ is false. The same logic works for the case A > $$\frac{b^3}{3}$$, just in reverse

Sorry and thank you for replying even though I may have been vague!

However, I do not understand how you get n that is greater than or equal to $$\frac{b^3}{A-\frac{b^3}{n}}$$

To the OP: if you want help from people who don't own the book, you can scan or photograph the book's proof and upload it here.

The book can be viewed here: http://www.scribd.com/doc/5874133/Calculus-Volume-1-686pp67

The proof starts from page 3 and ends at page 8. However, I am confused about the last bit, which is from pages 7-8.

If its hard to view on that, and you have bandwidth, please view the PDF file on the attachment.

#### Attachments

• acp1.pdf
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gom2 said:
The bit I am confused about is where his contradiction n$$\geq$$b$$^{3}$$/(A-b$$^{3}$$/n) came from?

gom2 said:
However, I do not understand how you get n that is greater than or equal to $\frac{b^3}{A-\frac{b^3}{n}}$
Well, firstly it is not n greater than or equal to what you wrote but rather n greater than or equal to (1) $$\frac{b^3}{A-\frac{b^3}{3}}$$. To figure out how to obtain this, remember that the inequality you obtained this from was valid for integer n ≥ 1.

Therefore, the inequality (2) n < $$\frac{b^3}{A-\frac{b^3}{3}}$$ is also valid for integer n ≥ 1.

However, the information on the right side of the inequality (2) is a constant. Thus, n can be made to be larger than the expression on the right. n can increase but the right side of (2) doesn't. n approaches infinity while the right side stays put. Obviously, n violates this inequality when (3) n ≥ $$\frac{b^3}{A-\frac{b^3}{3}}$$ (since the inequality (2) had n less than but not equal to the expression of (1)).
So there's a contradiction because the inequality (2) should be valid for all integer n ≥ 1, but as has been shown by (3), it is not valid for certain n. That's where the expression came from and is basically what Feldoh was getting at.

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## 1. How does Apostol's Archimedes area proof work for a parabola?

Apostol's Archimedes area proof for a parabola is based on the principle of exhaustion, which states that the area of a shape can be determined by approximating it with smaller, simpler shapes. In this proof, the parabola is broken down into an infinite number of infinitesimally small triangles, whose areas can be easily calculated. By adding up the areas of these triangles, the area of the parabola can be determined.

## 2. What makes Apostol's Archimedes area proof unique for parabolas?

Apostol's Archimedes area proof is unique because it uses a geometric approach rather than a calculus-based approach to determine the area of a parabola. This proof is also significant because it was one of the earliest known proofs for the area of a parabola, dating back to the 3rd century BC.

## 3. Can Apostol's Archimedes area proof be applied to other curves?

Yes, Apostol's Archimedes area proof can be applied to other curves as long as they can be broken down into infinitesimally small triangles and the area of each triangle can be easily calculated. This proof has been used to determine the areas of other conic sections, such as ellipses and hyperbolas.

## 4. What are the limitations of Apostol's Archimedes area proof?

Apostol's Archimedes area proof is limited in that it can only be used to determine the area of a parabola on a specific interval, usually from 0 to a certain value of x. It also requires a basic understanding of geometry and trigonometry to understand and apply the proof.

## 5. Why is Apostol's Archimedes area proof still relevant today?

Apostol's Archimedes area proof is still relevant today because it provides a fundamental understanding of the area under a curve, which is a key concept in calculus. It also serves as an example of the ingenuity and mathematical prowess of ancient mathematicians, and continues to inspire and inform modern mathematical research.

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