M408M Learning Module Pages
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Chapter 10: Parametric Equations
and Polar Coordinates
Learning module LM 10.1:
Parametrized Curves:
Learning module LM 10.2: Calculus with Parametrized Curves:
Slope and area
Arc length and surface area
Summary and simplification
Higher Derivatives
Learning module LM 10.3: Polar Coordinates:
Learning module LM 10.4: Areas and Lengths of Polar Curves:
Learning module LM 10.5: Conic Sections:
Learning module LM 10.6: Conic Sections in Polar Coordinates:
Chapter 12: Vectors and the Geometry of Space
Chapter 13: Vector Functions
Chapter 14: Partial Derivatives
Chapter 15: Multiple Integrals


Summary and simplification
Summary and Simplification
Our first set of results applies to all parametrized curves:
If we have a parametrized curve, where we know $x(t)$ and $y(t)$, then
The slope of the tangent line at time $t$ is
$\displaystyle{\frac{dy/dt}{dx/dt}}$.
The area under the curve from time $t_1$ to $t_2$ is $\displaystyle{\int_{t_1}^{t_2} y(t) \frac{dx(t)}{dt} dt.}$
The distance traveled along the curve from time $t_1$ to time $t_2$ is $\displaystyle{\int_{t_1}^{t_2} \sqrt{ \left ( \frac{dx}{dt}\right )^2 + \left ( \frac{dy}{dt} \right )^2}\; dt}$
The area of the surface of revolution, obtained by rotating around the $x$axis, is $$\displaystyle{\int_{t_1}^{t_2} 2 \pi y(t) \sqrt{ \left ( \frac{dx}{dt}\right )^2 + \left ( \frac{dy}{dt} \right )^2}\; dt}$$

If we happen to have a graph $y=f(x)$ then all of these formulas
simplify, since we can take the parametrization $x(t)=t$,
$y(t)=f(t)$. In that case $\frac{dx}{dt}=1$ and $\frac{dy}{dt} =
f'(t)=f'(x)$, so
The slope is $\displaystyle{\frac{dy/dt}{dx/dt} = \frac{f'(t)}{1} = f'(x)}$.
The area under the curve is $\displaystyle{\int_{t_1}^{t_2} f(t) dt = \int_{x_1}^{x_2} f(x) dx}$.
The arc length is $\displaystyle{\int_{t_1}^{t_2} \sqrt{1 + (f'(t))^2} dt =\int_{x_1}^{x_2} \sqrt{1 + (f'(x))^2} dx}.$
The surface area is $\displaystyle{\int_{t_1}^{t_2} 2 \pi f(t) \sqrt{1 + (f'(t))^2} dt =\int_{x_1}^{x_2} 2 \pi f(x) \sqrt{1 + (f'(x))^2} dx}.$

The first two are our usual formulas for slopes and areas. The third
and fourth may (or may not) be familiar to you from applications of
integration.
