If a smooth curve is given by the equation
in polar coordinates, then the arc length of the curve is expressed by the
integral:

where
and
are the values of the polar angle
at the endpoints of the arc
().

Find the length of the first turn of the spiral of Archimedes

**Solution.**

The first turn of the spiral is formed as the polar angle
changes from
to
.
Therefore

Find the length of the logarithmic spiral

between a certain point
and a moving point

**Solution**

In this case (no matter which of the magnitudes,
or
is greater!)

i.e., the length of the logarithmic spiral is proportional to the increment of
the polar radius of the arc.

Find the arc length of the cardioid

**Solution**

Here

Hence, by virtue of symmetry

Find the length of the closed curve

**Solution**

Since the function
is even, the given curve is symmetrical about the polar axis. Since the
function
{}
has a period of
during half the period from
to
{}
the polar radius increases from
to
and will describe half the curve by virtue of its symmetry. Further,

and

if
Hence,