Area, Volume, and Arc Length
Polar functions are a special case of parametric functions. These are just equations disguised as nasty trigonometric functions. It's sort of like they are 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.
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 are 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.