Arc Length and Surface Area with Parametric Equations

Arc length is simply how long a curve would be if you laid a string along it and then pulled the string straight. "Arc" here just means any piece of a curve, not only a piece of a circle. It's good for questions like how far a runner, a car or a point on a wheel actually traveled, and for how much cable or pipe a curved route needs. For a moving point, the length is how far it traveled, and the square root $\sqrt{(dx/dt)^2 + (dy/dt)^2}$ is just three small pieces: $dx/dt$ (how fast the point moves sideways), $dy/dt$ (how fast it moves up or down) and the square root (Pythagoras, combining those two into the actual speed along the curve). Multiply speed by a sliver of time $dt$ and add it all up; surface area just tacks on $2\pi y$, the distance around the circle each little piece sweeps out. It's less than it looks; bear with it.

Same idea as before, new variable. When you found arc length for $y = f(x)$, the key fact was the Pythagorean theorem on a tiny piece of curve: $ds = \sqrt{dx^2 + dy^2}$. That's still true here. The only change is that $x$ and $y$ now both move with $t$, so pull a $dt$ out of the square root:

Notice what's under the root. $\frac{dx}{dt}$ and $\frac{dy}{dt}$ are the point's horizontal and vertical speeds, and Pythagoras combines them into its actual speed along the curve. So $ds$ is speed times a sliver of time, and the integral is what your car's odometer does: add up speed $\times$ time to get distance traveled.

Traced once, or you count twice. Distance traveled and arc length are the same thing only if the point covers the curve once. Remember $x = \cos 2t$, $y = \sin 2t$ from the first lesson? On $0 \le t \le 2\pi$ it goes around the unit circle twice, so the integral gives $4\pi$, not $2\pi$. The odometer doesn't know you've been here before.

Surface area. Nothing new here either. Keep using $S = \int 2\pi r\,ds$, a thin band of circumference $2\pi r$ and width $ds$, with $r$ the distance to the axis: $y$ for the x-axis and $x$ for the y-axis.

A GPS watch computes the distance you ran by adding up speed × time along your parametric path. Cable-laying and pipe-routing software measures parametric curves the same way.