10.3 | Parametric Equations and Calculus


Parametric Form of the Derivative
If a smooth curve \(C\) is given by the equations
$$
x = f(t) \text{ and } y = g(t)
$$
the the slope of \(C\) at \((x,y)\) is
$$
\frac{dy}{dx} = \frac{dy/dt}{dx/dt}, \; \frac{dx}{dt} \not = 0.
$$
$$
\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx} \right)}{\frac{dx}{dt}}, \text{ provided } \frac{dx}{dt} \not = 0
$$
Let \(C\) be a smooth curve represented by the parametric equations \(x= f(t)\) and \(y = g(t)\) with parametric interval \([a,b]\). If \(C\) does not intersect itself, except possibly for \(t = a\) and \(t = b\), then the length of \(C\) is
$$
\begin{aligned}
L & = \int_a^b \sqrt{[f'(t)]^2 + [g'(t)]^2} \; dt \\
& = \int_a^b \sqrt{\left(\frac{dx}{dt} \right)^2 + \left(\frac{dy}{dt} \right)^2} \; dt
\end{aligned}
$$
Let \(C\) be a smooth curve represented by the parametric equations \(x = f(t)\) and \(y = g(t)\) with parameter interval \([a,b]\), and suppose that \(C\) does not intersect itself, except possibly for \(t = a\) and \(t = b\). If \(g(t) \geq 0\) for all \(t\) in \([a,b]\), then the area \(S\) of the surface obtained by revolving \(C\) about the \(x\)-axis is
$$
\begin{aligned}
S & = 2 \pi \int_a^b y \sqrt{[f'(t)]^2 + [g'(t)]^2} \; dt \\
& = 2 \pi \int_a^b y \sqrt{\left(\frac{dx}{dt} \right)^2 + \left(\frac{dy}{dt} \right)^2} \; dt
\end{aligned}
$$
If \(f(t) \geq 0\) for all \(t\) in \([a,b]\), then the area \(S\) of the surface obtained by revolving \(C\) about the \(y\)-axis is
$$
\begin{aligned}
S & = 2 \pi \int_a^b x \sqrt{[f'(t)]^2 + [g'(t)]^2} \; dt \\
& = 2 \pi \int_a^b x \sqrt{\left(\frac{dx}{dt} \right)^2 + \left(\frac{dy}{dt} \right)^2} \; dt
\end{aligned}
$$

E 10.3 Exercises