(PE4) Length of a Parametric Curve#
In this lesson, we are going to see how to calculate the length of a parametric curve.
Review Videos#
Arc Length#
Length of a Parametric Curve
If a curve \(C\) is described by the parametric equations
\[
x=f(t)\qquad y=g(t) \qquad \alpha \leq t \leq \beta
\]
then the length of \(C\) is:
\[
L = \int_{\alpha}^{\beta} \sqrt{\left(\dfrac{dx}{dt}\right)^2+\left(\dfrac{dy}{dt}\right)^2}\, dt
\]
Example 1#
Find the length of the parametric curve:
\[
x=3\cos t\qquad \quad y=3 \sin t \qquad 0\leq t\leq 2\pi
\]
Example 2#
Find the length of the parametric curve:
\[
x=3+t^4 \qquad \quad y=t^6 \qquad 0\leq t\leq 1
\]