integration of parametric functions
Many curves are most naturally described not as y = f(x) but by a parameter t that drives both coordinates at once: x = x(t), y = y(t). Think of a point tracing a cycloid or an ellipse as time runs. To find the area under such a curve, or its arc length, you do not first eliminate t — you integrate directly in t, letting the parameter do the bookkeeping for how fast the curve sweeps.
The key is the change-of-variable rule applied to the parameter. The area under a parametric curve is the integral of y dx, and since dx = (dx/dt) dt, this becomes the integral over the t-interval of y(t) times (dx/dt) dt. Arc length uses the speed: the length element ds = sqrt((dx/dt)^2 + (dy/dt)^2) dt, so total length is the integral of that speed over t. In each case you have simply rewritten the geometric quantity (a width dx, a length ds) in terms of dt using the chain rule.
Parametric integration is the natural setting for trajectories, for curves that loop or are not graphs of a single function (a circle fails the vertical-line test), and for any motion where time, not position, is the master variable. The standard pitfall is sign and orientation: when computing area swept, the direction the parameter runs determines the sign of (dx/dt), so a curve traced clockwise versus counterclockwise can flip the sign — exactly the orientation sensitivity that Green's theorem later makes precise.
For the ellipse x = a cos t, y = b sin t with t from 0 to 2pi, arc length is the integral of sqrt(a^2 sin^2 t + b^2 cos^2 t) dt — which has no elementary form and is in fact a complete elliptic integral.
Parametric arc length leads straight to the elliptic integral — which is precisely why that function was named after the ellipse.
Direction matters. The sign of dx/dt encodes which way the parameter sweeps, so for a closed loop the area integral changes sign with orientation — track it, do not take absolute values blindly.