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In order to simplify the problem, suppose we have a parabola $y=ax^2+bx+c$, here $a\neq0$, and a line $y=kx+d$, here $k\neq0$. We can assume that they will intersect at two different points. Thus, the $\Delta$ of the equation $ax^2+bx+c=kx+d$ will be greater than $0$($\Delta> 0$). Let $S$ be the area closed by them, it is clear that $S>0$.

Now I wonder how to calculate $S$ without calculus?



I try to solve this problem without calculus in order to make my little brother who don't know about calculus understand it. You can solve it as long as you can make a junior student understand your solution. :)

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Which is your meaning of area? –  Gastón Burrull Feb 16 '13 at 3:18
One can calculate without antiderivatives, in Riemann sum style, or in the Archimedean style, but limiting processes are inevitably involved. So in the wider sense of calculus, one cannot do it without calculus. –  André Nicolas Feb 16 '13 at 3:26
The same happens when we try to calculate the area of a circle. –  Gastón Burrull Feb 16 '13 at 3:29
@Gastón Burrull Edited. :) –  felix34 Feb 16 '13 at 3:32
@Gastón Burrull I don't understand. We can calculate the area of a circle by using the formula $S=\pi r^{2}$, isn't it? –  felix34 Feb 16 '13 at 3:46

1 Answer 1

up vote 3 down vote accepted

Archimedes derived a formula for this area. (It doesn't use calculus, which wouldn't be invented until centuries later.)

The area enclosed by a parabola (with vertical axis) and a chord AB is 4/3 the area of the triangle ABC where C is the point on the parabola whose x-coordinate is halfway between the x-coordinates of A and B. This result and Archimedes' method of deriving it are in the Wikipedia article The Quadrature of the Parabola.

I don't know if this will be any easier for your little brother to understand than a solution with calculus, but let us know.

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+1, This is very nice. I never knew that the area under a parabola could be computed by summing a geometric series. –  Eric Naslund Feb 16 '13 at 4:53
It's calculus, uses limits. –  Gastón Burrull Feb 16 '13 at 5:07
@GastónBurrull: While the question stated is "Calculate the area without calculus," which as you point out is impossible, I believe that the OP meant calculate the area without using Riemann sums. –  Eric Naslund Feb 16 '13 at 5:35

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