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# Parabolic Segment

The arc length of the parabolic segment

 (1)

illustrated above is given by

 (2) (3) (4)

and the area is given by

 (5) (6)

(Kern and Bland 1948, p. 4). The weighted mean of is

 (7) (8)

so the geometric centroid is then given by

 (9) (10)

The area of the cut-off parabolic segment contained between the curves

 (11) (12)

can be found by eliminating ,

 (13)

so the points of intersection are

 (14)

with corresponding -coordinates . The area is therefore given by

 (15) (16) (17)

The maximum area of a triangle inscribed in this segment will have two of its polygon vertices at the intersections and , and the third at a point to be determined. From the general equation for a triangle, the area of the inscribed triangle is given by the determinant equation

 (18)

Plugging in and using gives

 (19)

To find the maximum area, differentiable with respect to and set to 0 to obtain

 (20)

so

 (21)

Plugging (21) into (19) then gives

 (22)

This leads to the result known to Archimedes in the third century BC, namely

 (23)

Circular Segment, Geometric Centroid, Parabola

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## References

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 125, 1987.Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 4, 1948.

## Cite this as:

Weisstein, Eric W. "Parabolic Segment." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ParabolicSegment.html