We now need to move into the Calculus II applications of
integrals and how we do them in terms of polar coordinates. In this section we’ll look at the arc length
of the curve given by,
where we also assume that the curve is traced out exactly
once. Just as we did with the tangent lines in polar coordinates we’ll first
write the curve in terms of a set of parametric equations,
and we can now use the parametric formula for finding the
arc length.
We’ll need the following derivatives for these computations.
We’ll need the following for our ds.
The arc length formula for polar coordinates is then,
Let’s work a quick example of this.
Example 1 Determine
the length of .
Solution
Okay, let’s just jump straight into the formula since this
is a fairly simple function.
We’ll need to use a trig substitution here.
The arc length is then,

Just as an aside before we leave this chapter. The polar equation is the equation of a spiral. Here is a quick sketch of for .