Arc Length in Polar Coordinates

If a smooth curve is given by the equation $r=r(\theta )$ in polar coordinates, then the arc length of the curve is expressed by the integral:
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where $\alpha $ and $\beta $ are the values of the polar angle $\theta $ at the endpoints of the arc ($\alpha <\beta $).

Example 1

Find the length of the first turn of the spiral of Archimedes $r=a\theta .$

Solution.

The first turn of the spiral is formed as the polar angle $\theta $ changes from $0$ to $2\pi $. Therefore
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Example 2

Find the length of the logarithmic spiral
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between a certain point MATH and a moving point $(r,\theta ).$

Solution

In this case (no matter which of the magnitudes, $r$ or $r_{0}$ is greater!)
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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
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Solution

Here
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Hence, by virtue of symmetry
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Example 4

Find the length of the closed curve
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Solution

Since the function MATH is even, the given curve is symmetrical about the polar axis. Since the function MATH{} has a period of $4\pi ,$ during half the period from $0$ to $2\pi ${} the polar radius increases from $0$ to $a,$ and will describe half the curve by virtue of its symmetry. Further,
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and
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if MATH Hence,
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