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Define polar coordinates.

A system using radius (r) and angle (θ) to locate points.

What is a polar curve?

A curve defined by an equation in polar coordinates, typically r = f(θ).

Define a sector in the context of polar coordinates.

A 'slice' of a circle defined by an angle θ and radius r.

What is a limacon?

A polar curve described by the equation $r = a + bsin( heta)$ or $r = a + bcos( heta)$ .

Define the area element in polar coordinates.

Infinitesimal area element used in integration, given by $\frac{1}{2}r^2 d\theta$ .

Steps to find area inside $r = 2\cos(\theta)$ from $0$ to $\pi$ ?

Set up: $A = \frac{1}{2}\int_{0}^{\pi}(2\cos(\theta))^2 d\theta$ . 2. Simplify. 3. Integrate and evaluate.

Steps to find area of one petal of $r = \sin(2\theta)$ ?

Find range for one petal (e.g., $0$ to $\frac{\pi}{2}$ ). 2. Set up integral: $A = \frac{1}{2}\int_{0}^{\frac{\pi}{2}}(\sin(2\theta))^2 d\theta$ . 3. Integrate.

How to find the area of a polar curve when symmetry is present?

Identify symmetry. 2. Find the limits of integration for one symmetrical part. 3. Integrate and multiply by the appropriate factor.

Steps to calculate area enclosed by $r=3+3\sin(\theta)$ ?

Sketch. 2. Recognize symmetry. 3. Set up integral: $A = \frac{1}{2}\int_{0}^{2\pi}(3+3\sin(\theta))^2 d\theta$ . 4. Simplify and solve.

How to deal with $\sin^2(\theta)$ or $\cos^2(\theta)$ in polar area integrals?

Use the half-angle identities: $\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$ and $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$ .

What is the first step in finding the area enclosed by a polar curve?

Sketch the curve to understand its shape and any symmetries.

Area of a sector in polar coordinates?

$A = \frac{1}{2}r^2\theta$

Area enclosed by polar curve $r = f(\theta)$ from $a$ to $b$ ?

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

Half-angle identity for $\cos^2(\theta)$ ?

$\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}$

Area of one petal of a rose curve?

$A = \frac{1}{2}\int_{a}^{b} [f(\theta)]^2 d\theta$ , where the limits a and b define one petal.