Question

Find the arc length function for the curve with starting point (0,1). y=sin−1x+√1−x2

Answer 1

dy/dx = 1/√(1 - x^2) - x/√(1 - x^2) = (1 - x)/√(1 - x^2) 

1 + (dy/dx)^2 = 1 + (1 - x)^2/(1 - x^2) = [(1 - x^2) + (1 - x)^2]/(1 - x^2) = (1 - x^2 + 1 - 2x + x^2)/(1 - x^2) = (2 - 2x)/(1 - x^2) = 2(1 - x)/[(1 + x)(1 - x)] = 2/(1 + x) 

ds = √(1 + (dy/dx)^2 = √(2/(1 + x)) = √2/√(1 + x) dx 

L = √2 ∫ dx/√(1 + x) {0, 1} = √2 ∫ (1 + x)^(-1/2) dx = 2√2 √(1 + x) {0, 1} = 2√2 (√2 - 1) = 4 - 2√2 ≈ 1.1716

The arc length would be 1.1716 units.

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