Parabolic Segment Area (Archimedes Theorem)

Find the area of a parabolic segment bounded by a chord and a parabola from the chord width (base) and height. The area equals two thirds of base times height, with ratio and arc length too.

Input

Calculate the area of a parabolic segment bounded by a parabola and its chord from the chord width (base) and the height.

Chord width (base) b

Height h

Result

Segment area

Enclosing triangle area

Segment to rectangle ratio

2 : 3

Parabolic arc length

15.073706

By the Archimedes theorem the segment area equals (2/3) times base times height.

Lengths use the same unit as the input, and area is in that unit squared.

How it works

A region bounded by a parabola and a chord across it is called a parabolic segment.
By the quadrature of the parabola due to Archimedes, the segment area equals four thirds of the triangle that shares the same base and height.
With chord width as the base b and the distance from the vertex to the chord as the height h, the segment area is (2/3) times b times h.
The enclosing triangle has area (1/2) times b times h, so the triangle to segment area ratio is 3 to 4 and the segment to bounding rectangle ratio is 2 to 3.
The arc length is the length of the parabolic arc, computed from a closed form obtained by placing the vertex at the origin as y equals a times x squared and integrating.
Length units match the input, and area is in those units squared.

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