How long is a curve from x = a to x = b? Pythagoras over a tiny step, integrated.
Try this
1.5
length of y = 2x from x = 0 to x = b: L = ∫₀ᵇ √(1 + 2²) dx = √5 · b ≈ 2.236 b = 3.35
Pythagoras, sliced thin
Over a tiny step dx the curve rises by dy = f′(x) dx, so the little hypotenuse is √(dx² + dy²) = √(1 + [f′(x)]²) dx. Add up infinitely many of those tiny hypotenuses — that's the integral in the formula above.
Arc-length integrals are notoriously ugly — that √(1 + (f′)²) rarely has an elementary antiderivative. Even y = x² leads to a messy result, so numerical methods do a lot of the heavy lifting here.
Your turn
What's the arc length of the straight line y = 3x from x = 0 to x = 4 (using the formula)?
Arc length