Parametric, polar and vector valued functions
A path that loops and crosses itself is not the graph of any function. How do we find its slope, its length, the speed along it and the area it encloses?
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What this area is about
Single variable calculus works on graphs y = f(x). Many curves are not graphs: a seat on a spinning ride, a spirograph star, a spiral shell. They are described instead by a parameter, as x(t) and y(t), by an angle, as r = f(\theta), or by a position vector \mathbf{r}(t) = \langle x(t), y(t) \rangle. This area carries derivatives and integrals over to all three.
How these fit together
Parametric curves. Derivatives of parametric curves gives the slope \displaystyle \frac{dy}{dx} = \frac{dy/dt}{dx/dt} wherever \displaystyle \frac{dx}{dt} \ne 0, with tangent lines and concavity. Arc length and speed takes the speed \displaystyle \sqrt{x'(t)^2 + y'(t)^2} from the same two rates, for x' and y' continuous on [a, b], and integrates it to get the distance traveled, which is the length of the curve when it is traced once.
Polar curves. Slope of a polar curve treats r = f(\theta) as the parametric curve x = f(\theta)\cos\theta, y = f(\theta)\sin\theta. Area in polar coordinates adds thin wedges, \displaystyle \frac{1}{2}r^2\,d\theta, and subtracts an inner curve from an outer one.
Vectors and motion. Vector valued functions and motion in the plane makes velocity a vector, speed its length, and distance the integral of speed.
A path through it
Start with parametric derivatives. Slope of a polar curve and arc length both build on it, and motion in the plane comes after arc length. Polar area stands on its own and can come at any point.
Next come vector valued functions in space and double integrals in polar coordinates.