Arc Length in Polar Coordinates

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 ().

Example 1

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

Example 2

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.

Example 3

Find the arc length of the cardioid

Solution

Here

Hence, by virtue of symmetry

Example 4

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,

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