Question

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

Accepted Answer

First, understand the problem: we need to find the area of the region that lies inside the limaçon given by $$r = 2 + \cos	heta$$ and outside the circle given by $$r = 2. $$This means we want the area between these two curves. Set up the integral for the area between two polar curves. The general formula for the area between two curves $$r = r_1$(	heta)$$ and $$r = r_2$(	heta)$$ from $$	heta = a to$$ $$	heta = b is$$:

$$
	ext{Area} = \frac{1}{2} \int_a^b \left( r_1(	heta)^2 - r_2(	heta)^2 ight) d	heta
$$

Here, $$r_1$$(	heta)$$ is the outer curve and r_2$(	heta) is $$the inner curve. Determine the points of intersection between the two curves by setting $$2 + \cos	heta = 2. $$Solve for $$	heta$$:

$$
2 + \cos	heta = 2 \implies \cos	heta = 0
$$

Find the values of $$	heta in$$ $$[0, 2\pi]$$ where this holds true. Identify the correct interval(s) for $$	heta$$ over which the limaçon is outside the circle. Since $$\cos	heta = 0 at$$ $$	heta = \frac{\pi}{2}$$ and $$	heta = \frac{3\pi}{2}$$, these will be the limits of integration for the region where the limaçon is outside the circle. Set up the definite integral for the area:

$$
	ext{Area} = \frac{1}{2} \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \left( (2 + \cos	heta)^2 - 2^2 ight) d	heta
$$

This integral will give the area of the region inside the limaçon and outside the circle.

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

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Concetti chiave

Polar Coordinates and Graphing

Area Calculation in Polar Coordinates

Determining Intersection Points and Limits of Integration