It is known that if a smooth curve can be described using the function

     y = f(x) for a ≤ x ≤ b,

then the arc length is defined as

The arc length for a curve described using parametric equations will be introduced in this section.

Arc Length: Parametric Form

Differential Arc Length

Example: Arc Length of a Circle

When a curve is given using the following parametric equations

     x = f(t) and y = g(t) for a ≤ t ≤ b,

the arc length of the curve is defined as

where ds is the differential arc length. The expression for the differential arc length is

Note that the definition of the arc length is based on the assumption that both dx/dt and dy/dt are continuous. The arc length is basically the sum of the differential arc length ds along the curve.

Consider a circle with its center at (0,0) and a radius of 2 inches given by the following parametric equations as shown in the figure.

     x = 2cos(t) and y = 2sin(t) for 0 ≤ t ≤ 2π

According to the definition of the arc length, the circumference of the circle is given by: