Arc Length (Parametric)

Unit 9 · Parametric, Polar & Vector

Arc Length of Parametric Curves

The arc length of a parametric curve from t₁ to t₂ is L = ∫√((dx/dt)² + (dy/dt)²) dt. The integrand equals the speed |v(t)| of a particle, so arc length equals total distance traveled (provided the curve is traced once).

Arc Length (Parametric)

L = \int_{t_1}^{t_2}\sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\,dt

Speed = integrand = |v(t)|; total distance = ∫|v(t)| dt

AP Tip: After squaring and adding the derivatives, look for a perfect square under the radical — this simplifies the integral significantly.

Type 1

Computing Parametric Arc Length

Compute dx/dt and dy/dt, form the integrand, simplify, then integrate.

Example 1

Find the arc length of x = 3cos t, y = 3sin t on [0, π].

Example 2

Find the arc length of x = t², y = t³ from t = 0 to t = 1.

Type 2

Arc Length and Speed

Recognize that arc length equals total distance traveled, using speed as the integrand.

Example 3

x(t) = cos t, y(t) = sin t. Find total distance traveled from t = 0 to t = 2π.

Example 4

dx/dt = 3 and dy/dt = 4 (constant) on [0, 5]. Find the arc length.

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