TOPICS
Search

Parabolic Segment


ParabolicSegment

The arc length of the parabolic segment

 y=h(1-(x^2)/(a^2))
(1)

illustrated above is given by

s=int_(-a)^asqrt(1+y^('2))dx
(2)
=2int_0^asqrt(1+y^('2))dx
(3)
=sqrt(a^2+4h^2)+(a^2)/(2h)sinh^(-1)((2h)/a),
(4)

and the area is given by

A=int_(-a)^ah(1-(x^2)/(a^2))dx
(5)
=4/3ah
(6)

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

<y>=int_(-a)^aint_0^(h(1-x^2/a^2))ydxdy
(7)
=8/(15)ah^2,
(8)

so the geometric centroid is then given by

y^_=(<y>)/A
(9)
=2/5h.
(10)
ParabolicSegment2

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

y=x^2
(11)
y=ax+b
(12)

can be found by eliminating y,

 x^2-ax-b=0,
(13)

so the points of intersection are

 x_+/-=1/2(a+/-sqrt(a^2+4b)),
(14)

with corresponding y-coordinates y_+/-=x_+/-^2. The area is therefore given by

A=int_(a-sqrt(a^2+4b))^(a+sqrt(a^2+4b))[(ax+b)-x^2]dx
(15)
=1/6(a^2+4b)sqrt(a^2+4b)
(16)
=1/6(a^2+4b)^(3/2).
(17)

The maximum area of a triangle inscribed in this segment will have two of its polygon vertices at the intersections (x_-,y_-) and (x_+,y_+), and the third at a point (x^*,y^*) to be determined. From the general equation for a triangle, the area of the inscribed triangle is given by the determinant equation

 A_Delta=|x^- y^- 1; x^+ y^+ 1; x^* y^* 1|.
(18)

Plugging in and using y_*=x_*^2 gives

 A_Delta=1/2[b+(a-x^*)x^*]sqrt(a^2+4b).
(19)

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

 (partialA_Delta)/(partialx_*)=1/2(a-2x^*)sqrt(a^2+4b)=0,
(20)

so

 x_*=1/2a.
(21)

Plugging (21) into (19) then gives

 A=1/8(a^2+4b)^(3/2).
(22)

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

 A/(A_Delta)=(1/6)/(1/8)=4/3.
(23)

See also

Circular Segment, Geometric Centroid, Parabola

Explore with Wolfram|Alpha

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

Subject classifications