45–60. Areas of regions Find the area of the following regions.


The region inside the limaçon r = 4 - 2 cos θ

Verified step by step guidance

1. Recall that the area enclosed by a polar curve \( r = f(\theta) \) from \( \theta = a \) to \( \theta = b \) is given by the formula: \[ A = \frac{1}{2} \int_{a}^{b} r^2 \, d\theta \]
2. Identify the curve given: \( r = 4 - 2 \cos \theta \). Since this is a limaçon, and the problem asks for the area inside the entire curve, we consider \( \theta \) from 0 to \( 2\pi \) to cover the full region.
3. Set up the integral for the area using the formula: \[ A = \frac{1}{2} \int_{0}^{2\pi} (4 - 2 \cos \theta)^2 \, d\theta \]
4. Expand the square inside the integral to simplify the integrand: \[ (4 - 2 \cos \theta)^2 = 16 - 16 \cos \theta + 4 \cos^2 \theta \]
5. Split the integral into separate terms and use known integral formulas for \( \cos \theta \) and \( \cos^2 \theta \) over the interval \( [0, 2\pi] \) to evaluate each part before combining them to find the total area.

Key Concepts

Polar Coordinates and Graphing

Polar coordinates represent points using a radius and an angle, expressed as (r, θ). Understanding how to graph polar equations like r = 4 - 2 cos θ helps visualize the region whose area is to be found, especially for curves like limaçons that have distinctive shapes.

Area Calculation in Polar Coordinates

The area enclosed by a polar curve r(θ) from θ = a to θ = b is given by (1/2) ∫ from a to b of [r(θ)]² dθ. This formula is essential for finding the area inside curves defined in polar form, requiring setting correct integration limits and squaring the radius function.

Properties of Limaçon Curves

Limaçons are a family of polar curves defined by equations like r = a ± b cos θ. Their shape varies depending on the relationship between a and b, which affects whether the curve has an inner loop, dimple, or is convex. Recognizing these properties aids in determining integration bounds and interpreting the region.