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Arc length is the actual distance along a curve between two points — not the straight-line distance. For a smooth curve y = f(x) from x = a to x = b, the arc length formula is: L = ∫[a to b] √(1 + [f'(x)]²) dx.
The formula comes from summing infinitely many tiny straight-line segments along the curve. Each segment has length √(dx² + dy²) = √(1 + (dy/dx)²) dx. Integrating gives the exact total length.
For parametric curves x = x(t), y = y(t) from t = α to t = β: L = ∫[α to β] √((dx/dt)² + (dy/dt)²) dt. For polar curves r = f(θ): L = ∫[α to β] √(r² + (dr/dθ)²) dθ.
Arc length integrals often cannot be solved in closed form — numerical methods (Simpson's rule, Gaussian quadrature) are used in practice. When they are solvable, the key step is simplifying √(1 + [f'(x)]²) into a perfect square.
L = ∫[a→b] √(1 + [f'(x)]²) dx
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f(x) = x (straight line)
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[0, 3]
Arc Length Formulas
y = f(x): L = ∫[a→b] √(1 + [f'(x)]²) dx
Parametric: L = ∫[α→β] √((x')² + (y')²) dt
Polar: L = ∫[α→β] √(r² + (dr/dθ)²) dθ
This calculator is for educational purposes only and does not constitute professional advice. Results are based on standard mathematical formulas. Always verify critical calculations with a qualified professional before making important decisions.