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Hyperbolic Segment Area Calculator

Find the area of a hyperbolic segment bounded by x²/a²−y²/b²=1 and a vertical chord, computed by numerical integration from a, b and the cut position x.

Input

Find the area of the segment bounded by the right branch of x²/a²−y²/b²=1 and a vertical chord at the cut position x.

The vertex is at x=a, and the height at the cut position x is y₀ = b·√(x²/a²−1).

Outside the vertex, greater than a

Result

VertexChord

Segment area

7.901763

Semi-axis a

3

Semi-axis b

2

Cut position x

6

Half height y₀

3.464102

Chord length

6.928203

The area is in squared length units. Use the same unit for a, b and x.

How it works

  • The hyperbola is given by x²/a²−y²/b²=1, with its vertex at x=a. At the cut position x the curve height is y₀ = b·√(x²/a²−1), so the chord length is twice that value, 2y₀.
  • The segment area equals the rectangular strip from the vertex a to the chord x minus the region under the curve, summed for the upper and lower halves. In formula form A = 2·∫[a..x] (y₀ − b·√(t²/a²−1)) dt.
  • This integral is evaluated by numerical integration using the composite Simpson rule. The interval is divided finely enough to give a high-accuracy area for practical use.
  • The cut position x must lie outside the vertex, meaning it must be greater than a. For x at or below a the chord does not meet the branch, so no area is defined.
  • The area is expressed in squared length units. Enter a, b and x in the same unit so the result stays consistent.

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