From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!
We have changed our privacy policy. In addition, we use cookies on our website for various purposes. By continuing on our website, you consent to our use of cookies. You can learn about our practices by reading our privacy policy.
© 2016 Shmoop University, Inc. All rights reserved.
Area, Volume, and Arc Length

Area, Volume, and Arc Length

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).

Answer.

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.

People who Shmooped this also Shmooped...

Advertisement