# Area of a Parabola

In my unending quest to find out seemingly simple bits of maths that I didn’t know, one of my year 10 students found this:

From http://www.tasonline.co.za/toolbox/area/parabola.htm

So three questions:

1. Is this true?

2. Is this useful?

3. Is it better/easier than integrating?

If it is true, I think it falls into the “wow that’s neat” category.

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Yep, it’s totally truestart withf(x) = (b/2 – x)(b/2 +x)*c = (b^2*c/4 – c*x^2)h = b^2*c/4 => c = h*4/b^2sof(x) = h-h*4*x^2/b^2solve the integral from 0 to b/2 and multiply by 2 and you gethb*2/3pretty sexy actually.

I think appolonius may have a geometric proof of this as well.

Yes. It’s even true if the parabolic segment is not symmetric. This is (more or less) how Archimedes described the quadrature of the parabola.http://en.wikipedia.org/wiki/The_Quadrature_of_the_ParabolaI have used it occasionally, and it can be easier than integrating.

dave, As daveinstpaul pointed out, Archimedes did this, and he did it with perhaps the first use of summing an infinite geometric series in math history.