44–49. Areas of regions Find the area of the following regions.
The region inside the limaçon r=2+cosθ and outside the circle r=2

Verified step by step guidance

1. First, understand the problem: we need to find the area of the region that lies inside the limaçon given by \(r = 2 + \cos\theta\) and outside the circle given by \(r = 2\). This means we want the area between these two curves.
2. Set up the integral for the area between two polar curves. The general formula for the area between two curves \(r = r_1(\theta)\) and \(r = r_2(\theta)\) from \(\theta = a\) to \(\theta = b\) is: \[ \text{Area} = \frac{1}{2} \int_a^b \left( r_1(\theta)^2 - r_2(\theta)^2 \right) d\theta \] Here, \(r_1(\theta)\) is the outer curve and \(r_2(\theta)\) is the inner curve.
3. Determine the points of intersection between the two curves by setting \$2 + \cos\theta = 2\(. Solve for \)\theta\(: \[ 2 + \cos\theta = 2 \implies \cos\theta = 0 \] Find the values of \)\theta\( in \)[0, 2\pi]$ where this holds true.
4. Identify the correct interval(s) for \(\theta\) over which the limaçon is outside the circle. Since \(\cos\theta = 0\) at \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\), these will be the limits of integration for the region where the limaçon is outside the circle.
5. Set up the definite integral for the area: \[ \text{Area} = \frac{1}{2} \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \left( (2 + \cos\theta)^2 - 2^2 \right) d\theta \] This integral will give the area of the region inside the limaçon and outside the circle.

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 = 2 + cosθ (a limaçon) and r = 2 (a circle) is essential to visualize the regions whose areas are to be found.

Area Calculation in Polar Coordinates

The area enclosed by a polar curve r(θ) between angles α and β is given by (1/2) ∫ from α to β of [r(θ)]² dθ. This formula is fundamental for finding areas bounded by one or more polar curves.

Determining Intersection Points and Limits of Integration

To find the area between two polar curves, it is necessary to find their points of intersection by solving r1(θ) = r2(θ). These intersection angles define the limits of integration for the area calculation.