Determine an expression for the length of the curve $r = f(\theta)$ between $\theta = a$ and $\theta = b$.
I think I will need to convert to rectangular coordinates in some way. After that, I will probably need to find an integral to express the length. How do I go about doing this?
There is a similarity between the expressions in Cartesian and polar coordinates.
In both expressions, you are finding the length of the arc by combining motion in two perpendicular coordinates.
In rectangular, you have the x-speed, which is $1$, and the y-speed, which is $\displaystyle \frac{dy}{dx}$ (per x unit), hence $\displaystyle \int \sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx$
In polar, however, the curve moves in the tangential direction with speed $f(\theta)$ (per radian) and in the outward direction with speed $f'(\theta)$ per radian, hence $\displaystyle \int \sqrt{f(\theta)^2+f'(\theta)^2}d\theta$.
Parametric is the most obvious, in terms of $t$, it is simply $\displaystyle \int \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dx$ using both the $x$ velocities and $y$ velocities at each point.