Ximpledu

Length of a Curve

See how to find the length of a curve
(y = f(x), parametric function).
2 examples and their solutions.

Length of a Curve: y = f(x)

Formula


l = ab1 + [f'(x)]2 dx
Proof
Definite Integral

Example

Find the length of the given curve from x = 1 to x = e.

y = 12x2 - 14 ln x
Solution

Length of a Curve: Parametric Function

Formula


l = ab(dxdt)2 + (dydt)2 dt
Proof
Definite Integral

Example

Find the length of the given curve from t = 0 to t = 2π.

x = t - sin t
y = 1 - cos t
Solution