Arc length of a parametric curve

 Let's consider the plane curve defined by the following parametric equations:

x = x(t) y = y(t)    t₁≤t ≤t₂ and let's assume that x(t) and y(t) are differentiable functions of t, then the arc length of the parametric curve is given by:

This video gives an idea of where this formula originates:

 

Example

Find the arc length of the semicircle defined by the equations:

Solution

Here is the graph of the semicircle:

Let's use the formula to find the arc length:

Practice

Find the arc length of the curve defined by the equations: