Arc Length of Curves – AP Calculus BC

       CONTINUOUS LENGTH FORMULA

       $L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} dx$

Horizontal $dx$
Vertical $dy$
Hypotenuse $\sqrt{dx^2 + dy^2}$

Select Path Curve

Discretization Resolution

Segments $n$ = 5

DISCRETE APPROX OVER $[0, 3]$

Hypotenuses Sum: 4.500

TRUE THEORETICAL INTEGRAL

Calculus Length $L =$ 5.000

Error Diff: 0.500

The Limit: As $n \to \infty$, the straight $ds$ line segments perfectly mold out the curve. Because a straight line is the shortest distance between two points, any finite $n$ approximation will ALWAYS be an UNDERESTIMATE of the true bent arc length!