What is a polar curve?
A curve defined using polar coordinates $(r, \theta)$, where $r$ is the distance from the origin and $\theta$ is the angle from the positive x-axis.
Define polar coordinates.
A two-dimensional coordinate system where each point is determined by a distance $r$ from a reference point (the pole) and an angle $\theta$ from a reference direction (the polar axis).
What does $r$ represent in polar coordinates?
$r$ represents the radial distance from the origin to the point in the polar coordinate system.
What does $\theta$ represent in polar coordinates?
$\theta$ represents the angle measured counterclockwise from the positive x-axis to the line segment connecting the origin to the point.
How do you find the intersection points of two polar curves?
Set the two polar equations equal to each other, $r_1(\theta) = r_2(\theta)$, and solve for $\theta$. These $\theta$ values are the angles at the intersection points.
Outline the steps to find the area between $r = 3$ and $r = 3 - 2\sin(2\theta)$ in the second quadrant.
1. Identify $r_1 = 3$ and $r_2 = 3 - 2\sin(2\theta)$. 2. Determine bounds: $\frac{\pi}{2}$ to $\pi$. 3. Set up the integral: $A = \frac{1}{2} \int_{\frac{\pi}{2}}^{\pi} [(3 - 2\sin(2\theta))^2 - 3^2] d\theta$. 4. Evaluate the integral.
How do you determine which polar function is the 'inner' function?
Imagine yourself at the origin. The function you see first is the inner function.
What is the formula for the area of a region bounded by a single polar curve?
$A = \frac{1}{2} \int_a^b r^2 d\theta$
What is the formula for the area of a region bounded by two polar curves?
$A = \frac{1}{2} \int_a^b (r_2^2 - r_1^2) d\theta$, where $r_2$ is the outer radius and $r_1$ is the inner radius.
How do you find the area between two polar curves?
$A = \frac{1}{2} \int_{\alpha}^{\beta} [f(\theta)^2 - g(\theta)^2] d\theta$, where $f(\theta)$ is the outer curve and $g(\theta)$ is the inner curve.
