Shortcut to the area under a curve?

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Discussion Overview

The discussion revolves around finding the area under a curve defined by a second-degree polynomial, particularly focusing on the area between the roots of the polynomial. Participants explore potential methods to calculate this area without relying on calculus, while also referencing historical approaches such as Archimedes' method of exhaustion.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant inquires about an easy method to calculate the area under a curve described by a second-degree polynomial, specifically between its zeros.
  • Another participant references Archimedes' method of exhaustion as a historical approach to finding areas under parabolas, noting its reliance on series methods.
  • A suggestion is made that if one can determine the area under the curve of 1-x², it could be used to find areas under other parabolas through transformations.
  • There is a claim that there is a known solution to the area under 1-x², which can be easily found using calculus, although the exact area is debated.
  • A formula for the area under the curve a(x-b)(x-c) is presented, but it is noted that deriving this may require calculus.
  • One participant expresses uncertainty about the existence of an exact solution for the area under 1-x², prompting clarification from others.
  • Another participant emphasizes the simplicity of Archimedes' method, suggesting it relies on basic geometry rather than segmentation or summation.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of calculus for finding areas under specific curves, with some asserting that known solutions exist while others remain uncertain. The discussion does not reach a consensus on the best approach or the existence of simpler methods.

Contextual Notes

Some participants acknowledge that their understanding and methods may depend on calculus, while others explore historical methods that do not. There are unresolved questions about the exact area under certain curves and the applicability of various methods.

kenewbie
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I'm not up to calculus yet, but I was playing with functions and I had this idea that I wanted to ask about.

Is there an easy way to get the area under a curve given these restrictions:

1. The curve is described by second degree polynomial
2. The area we are interested in is the complete range between the zero's on the x-axis

In other words, what is the area under the curve between the zero's of a(x-b)(x-c)

It seems to be that this should be narrow enough to be calculable without doing series and integrals and whatnot (calculus, I don't know the exact approach yet), but I can't seem to find anything that fits.

So, is there a nice solution to this?
 
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If you had a way to find the area under the specific curve 1-x2 then you could find the area under any other parabola by shifting and rescaling vertically and horizontally. You can do this conversion algebraically by completing the square and a change of variables.

Of course, the challenge remains to find the area under 1-x2.
 
Maze: That's some real nice insight, I didn't even consider that. You are 100% correct of course, no need to worry about a, b, and c but just focus on 1-x^2. Thanks.

From what you are saying I gather that there is no known exact solution to the area under 1-x^2?

HallsofIvy I've read some popular recounts of Archimedes' approach and yes that is segmentation and summation. My thought was that perhaps there are certain classes of curves that are easier to get the area of, like second degree polynomials with integer zeroes, for example.

k
 
Uh, no, there is a known solution to the area under 1-x^2. Calculus makes it pretty easy to find.

I think the method of exhaustion, previously mentioned, states that this area - bounded by 1 - x^2 and the x-axis - equals 2/3 of the area of the smallest rectangle that covers the region... or is it 1/3... anyway, you might want to look into this.
 
You have f(x) = a(x-b)(x-c). Choose b <= c.

The area under the curve between b and c is
[tex]a(b-c)^3/6[/tex]
 
CRGreathouse: That's exactly the sort of thing i was trying to "construct". Can you show how that was derived or would that require calculus?

k
 
Yes, CRGreathouse used calculus to produce a general formula that could then be used as a "shortcut" for specific cases!
 
HallsofIvy ok, thank you. I'll just leave this until I get to calculus I guess.

k
 
  • #10
I just read an explanation of Archimedes' method, and wanted to say "Wow!"

It is not really segmentation/summation at all, the idea that what is true for parts must be true for the sum is just a tiny part of the proof. I don't want to spoil how he did it but rather

recommend that everyone look into how he did it.

http://www.math.ubc.ca/~cass/archimedes/parabola.html

Basically using nothing but primary school geometry.

k
 

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