# Arc Length in Polar Coordinates

If a curve is given in polar coordinates , an integral for the length of the curve can be derived using the arc length formula for a parametric curve. Regard as the parameter. The parametric arc length formula becomes

Now and , so

Square and add, using the fact that :

Hence,

Note: As with other arc length computations, it's pretty easy to come up with polar curves which lead to integrals with non-elementary antiderivatives. In that case, the best you might be able to do is to approximate the integral using a calculator or a computer.

Example. Find the length of the curve from to .

The length is

Example. Find the length of the cardiod for to .

I'll do the antiderivative separately:

The length is

Example. Find the length of the polar curve for to .

The length is

I'll do the antiderivative separately:

So

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