Welcome to Sections 10.3 and 10.4!

In these sections, we'll be exploring polar coordinates and their applications in calculating areas and arc lengths. Get ready to expand your mathematical toolkit and gain a new perspective on geometry!

10.3: Polar Coordinates

Unlike Cartesian coordinates, which use directed distances from perpendicular axes, polar coordinates use a distance from the origin (pole) and an angle from the polar axis. A point $P$ in the plane is represented by an ordered pair $(r, \theta)$, where:

  • $r$ is the distance from the pole $O$ to the point $P$.
  • $\theta$ is the angle (usually in radians) between the polar axis and the line segment $OP$.

Remember that a positive angle is measured counterclockwise, and a negative angle is measured clockwise. Also, note that $(0, \theta)$ represents the pole for any value of $\theta$. We can also extend polar coordinates to include negative values of $r$; in this case, $(-r, \theta)$ lies on the same line as $(r, \theta)$ but on the opposite side of the pole.

Key Conversion Formulas:

If a point $P$ has Cartesian coordinates $(x, y)$ and polar coordinates $(r, \theta)$, then the conversion formulas are as follows:

  • $x = r \cos \theta$
  • $y = r \sin \theta$

And conversely:

  • $r^2 = x^2 + y^2$
  • $\tan \theta = \frac{y}{x}$

Important Note: When converting from Cartesian to polar coordinates, remember that $\theta$ is not uniquely determined by the equation $\tan \theta = \frac{y}{x}$. You must choose $\theta$ so that the point $(r, \theta)$ lies in the correct quadrant.

Polar Curves

A polar equation is an equation in the form $r = f(\theta)$ or $F(r, \theta) = 0$. The graph of a polar equation consists of all points $P$ that have at least one polar representation $(r, \theta)$ whose coordinates satisfy the equation. Common examples include circles ($r=a$) and cardioids ($r = 1 + \sin \theta$). Utilizing symmetry can simplify sketching polar curves.

To find the tangent line to a polar curve $r = f(\theta)$, consider $x = r \cos \theta = f(\theta)\cos \theta$ and $y = r \sin \theta = f(\theta) \sin \theta$. Then,

$$\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{\frac{dr}{d\theta} \sin \theta + r \cos \theta}{\frac{dr}{d\theta} \cos \theta - r \sin \theta}$$

10.4: Areas and Lengths in Polar Coordinates

Now, let's use our knowledge of polar coordinates to calculate areas and arc lengths.

Area

The area $A$ of a region bounded by the polar curve $r = f(\theta)$ and the rays $\theta = a$ and $\theta = b$ is given by the integral:

$$A = \frac{1}{2} \int_{a}^{b} [f(\theta)]^2 d\theta = \frac{1}{2} \int_{a}^{b} r^2 d\theta$$

Remember to choose the limits of integration $a$ and $b$ carefully to sweep out the desired region exactly once!

Arc Length

The length $L$ of a polar curve $r = f(\theta)$ from $\theta = a$ to $\theta = b$ is given by the integral:

$$L = \int_{a}^{b} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta$$

Don't be intimidated by the formula! It's derived from the arc length formula for parametric curves, with $x = r \cos \theta$ and $y = r \sin \theta$.

Keep practicing, and you'll master these concepts in no time! Good luck, Professor Baker's Math Class!