# The method of exhaustion for the area of a parabolic segment

• Forrest T
In summary, the conversation is about the method of exhaustion used in the proof of the area under a parabola in Calculus Volume I by Apostol. The author uses two equations to establish the proof, but then discards parts of them without explanation. The justification for this is not clear and the question remains as to why these parts were discarded in the first place. The conversation also discusses the use of upper and lower sums and their relationship to the base and subdivisions of a parabolic segment. Ultimately, although the author's proof can be followed, the reasoning behind discarding parts of the equations is still unclear.
Forrest T
When I started reading Calculus Volume I by Apostol, I didn't fully understand his description of the method of exhaustion. I noticed that Caltech and MIT don't even teach from this section of the book, and decided not to worry about it. I understand why everything in the proof is true, I just don't understand why Apostol discarded parts of some equations. Please enlighten me.
Here is the part of the proof of the area under a parabola:

The author establishes the following equations:

12 + 22 + $\cdots$ + (n - 1)2 = n3/3 - n2/2 + n/6
12 + 22 + $\cdots$ + n2 = n3/3 + n2/2 + n/6

Then it says: For our purposes, we do not need the exact expressions given in the right-hand members of [these two equations]. All we need are the two inequalities

12 + 22 + $\cdots$ + (n - 1)2 < n3/3 < 12 + 22 + $\cdots$ + n2

which are valid for every integer n $\geq1$. These inequalities can be deduced easily as consequences of [the two above equations], or they can be proved directly by induction.

If we multiply both inequalities by b3/n3 and make use of [the equations for the upper sum (Sn) and lower sum (sn) we observe

sn < b3/3 < Sn

for every n. The author then proves that b3/3 is the only number between sn and Sn (proving the theorem).

It seems to me as though the author arbitrarily discards part of the first equations, and I don't understand the justification for this. Apostol led me to believe he was deriving the theorem, as Archimedes would have, through the method of exhaustion, rather than writing an unwieldy proof of a basic formula. In other words, I don't know why Archimedes would have discarded those parts of the first two equations while deriving the theorem.
Can someone please explain this to me? Thanks!

You need to give some more information. What is b and what are sn and Sn?

A parabolic segment is the area bounded by the parabola y=x2, the line x=b, and the x-axis. So b is the length of the base of the parabolic segment. n is the number of subdivisions of the base (each subdivision is of width b/n). Sn is the upper sum (the sum of the areas of rectangles drawn above the parabola, as an approximation) and sn is the lower sum, the sum of the areas of the rectangles drawn below the parabola.

I am not sure what you are asking, but sn and Sn have the same limit as n becomes infinite.

I realize that and can follow the proof. I just don't understand why the ends of the first two equations were discarded.

## 1. What is the method of exhaustion for the area of a parabolic segment?

The method of exhaustion is a mathematical technique used to approximate the area of a parabolic segment by dividing it into smaller and simpler shapes, such as triangles or rectangles, whose areas can be easily calculated.

## 2. Who first developed the method of exhaustion?

The method of exhaustion was first developed by the ancient Greek mathematician Eudoxus of Cnidus in the 4th century BC.

## 3. How does the method of exhaustion work?

The method of exhaustion works by dividing a parabolic segment into smaller and simpler shapes, and then finding the sum of their areas. As the number of shapes increases, the sum of their areas approaches the actual area of the parabolic segment.

## 4. What is the difference between the method of exhaustion and the method of exhaustion for a circle?

The method of exhaustion for a parabolic segment uses straight lines to approximate the curved shape, while the method of exhaustion for a circle uses polygons to approximate the circular shape.

## 5. What are some real-world applications of the method of exhaustion?

The method of exhaustion has been used in various fields, such as engineering, physics, and economics, to approximate and solve problems involving curved shapes and areas, such as calculating the volume of a sphere or the area under a curved graph.

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