Polar Curves-2:
1. Derive an expression for arc length in Cartesian form.
Solution: Let P (r,θ) and Q (r δr, θ δθ) be two neighbouring point on the graph of the functions r f (θ). So that they are at length S and S δs from a fixed point A on the curve.
∴ PQ = (S δs) - s = δs
Draw Pn Perpendicular to OQ,
From Δ*OPN,
PN/OP = Sin δθ
i.e PN/r = Sin δθ or PN = r Sin δθ
and ON/OP = Cos δθ
i.e ON/r = Cos δθ (or) ON = r Cos δθ
When Q is very close to P, the length of the arc PQ are equal to δs as the length of the chord PQ.
In Δ*PQN,
(PQ)2 = (PN)2 (QN)2
But PN = r Sin δθ
And QN = OQ - ON
∴ = r δr - r
∴ = δr
And (PQ)2 = (PN)2 (QN)2
(δS)2 ( rδθ )2 (δr)2............................(5)
Hence This is Beta-Gamma Functions.