Step-by-step explanation:
Consider two points on a curve, (xᵢ, yᵢ) and (xᵢ₊₁, yᵢ₊₁). The distance between them can be found with distance formula:
d = √((xᵢ₊₁ − xᵢ)² + (yᵢ₊₁ − yᵢ)²)
d = √((Δx)² + (Δy)²)
Factor out Δx:
d = √(1 + (Δy/Δx)²) Δx
Adding up the distances from i=1 to i=n, the total arc length can be approximated as:
s ≈ ∑ᵢ₌₁ⁿ √(1 + (Δy/Δx)²) Δx
Taking the limit as n approaches infinity, we get the exact value of s:
s = lim(n→∞) ∑ᵢ₌₁ⁿ √(1 + (Δy/Δx)²) Δx
Which can be written as a definite integral:
s = ∫ₐᵇ √(1 + (dy/dx)²) dx
