Calculus II

A Guide to Parametric & Polar Coordinates

1. Parametric Curves

Instead of defining a path with $y$ in terms of $x$, we use a third variable, $t$ (think of it as time). A point's location is given by its coordinates $(x(t), y(t))$. This lets us trace complex paths, like the flight of a projectile or a path that crosses itself.

Finding the Cartesian Equation

To see the underlying path, we try to eliminate the parameter $t$.

  1. Solve one of the equations for $t$.
  2. Substitute that expression for $t$ into the other equation.
  3. For trig functions, use identities like $\sin^2(t) + \cos^2(t) = 1$.

2. Calculus with Parametric Curves

Even with this new way of defining curves, we can still find slopes and lengths. We just need to adapt our formulas.

Slope of the Tangent Line

The slope $\frac{dy}{dx}$ is found by the ratio of how $y$ changes with respect to $t$ and how $x$ changes with respect to $t$.

Slope: $$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$$

Arc Length

To find the length of a path from $t=a$ to $t=b$, we integrate the "speed" along the curve. This comes from a version of the Pythagorean theorem.

Arc Length: $$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$$

3. Polar Coordinates

Instead of a rectangular grid $(x, y)$, we navigate with a distance from the origin, $r$, and an angle, $\theta$. A point is given by $(r, \theta)$. This is perfect for describing things with circular or rotational symmetry.

Conversion Formulas

These are the formulas you need to translate between Cartesian and Polar coordinates.

From Polar to Cartesian:

$x = r \cos(\theta)$

$y = r \sin(\theta)$

From Cartesian to Polar:

$r^2 = x^2 + y^2 \implies r = \sqrt{x^2 + y^2}$

$\tan(\theta) = \frac{y}{x}$

Warning: When finding $\theta$ from $\tan(\theta) = y/x$, always check which quadrant your $(x,y)$ point is in to make sure you have the correct angle!

4. Areas and Lengths in Polar Coordinates

We can also measure area and length using polar coordinates. The key is to think in terms of wedge-shaped sectors instead of rectangles.

Area of a Polar Region

To find the area swept out by a polar curve $r = f(\theta)$ from angle $\alpha$ to $\beta$, we use the following integral.

Area: $$A = \frac{1}{2} \int_\alpha^\beta [r(\theta)]^2 \, d\theta$$

This is one of the most important formulas to remember for polar coordinates.

Arc Length in Polar

This formula looks a bit like the parametric version, but adapted for polar variables $r$ and $\theta$.

Arc Length: $$L = \int_\alpha^\beta \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta$$