Arc Length for Polar Functions

Polar functions are a special case of parametric functions. These are just equations disguised as nasty trigonometric functions. It's sort of like they're wearing intricate Halloween costumes, like a stick figure man costume made of black cloth and glow sticks. We can't tell who it is, but we can do our best to use our knowledge about a person to figure it out.

The polar function r = f(t) can be parametrized as

x(t) = r cos t = f(t) cos t

y(t) = r sin t = f(t) sin t

We're using t instead of θ so we can talk about  instead of , to be consistent with the earlier discussion of parametric functions. To find  and , we have to apply the product rule carefully.

Sample Problem

Find the arc length for one petal of the polar function r = sin(3t).

The function looks like

The first petal corresponds to the interval . We get

x(t) = r cos t = sin(3t)cos t

y(t) = r sin t = sin(3t)sin t.

Using the product rule gives us

We use the arc length formula and get

It looks complicated and drawn out, but we can always just use a calculator to solve the integral. If our teacher asks us to solve this one, they're looking to torture us.

Sometimes, a slightly modified costume simplifies the calculations. We can use the identity

sin2 t + cos2 t = 1

to get a nicer formula for the arc length of a polar function.

If we want to make the formula look more like it goes with polar functions, we can put θ in place of t:

This formula is nice to keep handy, but we probably don't need to memorize it unless we need to do a lot of polar arc length problems.