# 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}