Parametric Equations

A curve can be described by y = F(x)

We now discuss another method for describing a curve, let x & y be specified as continuous function of another variable t (called a parameter) by

$x=f(t)$ & $y=g(t)$

Each value of t determines a point (x, y)

(x, y) = (f(t), g(t))

Arc Length §

Computing the length of a curve defined by y = f(x)

$L={\sum }_a^b\sqrt{1+(\frac{dy}{dx}{)}^2}dx$

For a curve defined parametrically where x = f(t) and y = g(t)

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

then

$\sqrt{1+(\frac{dy}{dx}{)}^2}dx=\sqrt{1+(\frac{\frac{dy}{dt}}{\frac{dx}{dt}}{)}^2}dx=\sqrt{(\frac{dx}{dt}{)}^2+(\frac{dy}{dt}{)}^2}dt$

thus

$L={\sum }_{\alpha }^{\beta }\sqrt{(\frac{dx}{dt}{)}^2+(\frac{dy}{dt}{)}^2}dt$

where

$f(\alpha )=a$ & $f(\beta )=b$