Question

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.The region inside the limaçon r = 2 + cos θ

Accepted Answer

First, understand the problem: we need to find the area of the region inside the lima\c con given by the polar equation $$r = 2 + \cos 	heta. $$This means we are looking for the area enclosed by this curve for $$	heta$$ ranging from $$0 to$$ $$2\pi. $$Recall the formula for the area enclosed by a polar curve $$r(	heta)$$ between angles $$\alpha$$ and $$\beta is $$given by:

$$
	ext{Area} = \frac{1}{2} \int_{\alpha}^{\beta} r(	heta)^2 \, d	heta
 In $$this problem, since the lima\c con is a closed curve traced once as $$	heta$$ goes from $$0 to$$ $$2\pi$$, set $$\alpha = 0$$ and $$\beta = 2\pi. $$Substitute $$r(	heta) = 2 + \cos 	heta$$ into the formula to get:

$$
	ext{Area} = \frac{1}{2} \int_0^{2\pi} (2 + \cos 	heta)^2 \, d	heta
$$ Next, expand the square inside the integral:

$$
(2 + \cos 	heta)^2 = 4 + 4 \cos 	heta + \cos^2 	heta
  
So $$the integral becomes:

$$
\frac{1}{2} \int_0^{2\pi} \left(4 + 4 \cos 	heta + \cos^2 	hetaight) d	heta
$$ Finally, split the integral into three separate integrals and evaluate each one using known integral formulas for $$\cos 	heta$$ and $$\cos^2$ 	heta. $$Remember to use the identity for $$\cos^2$ 	heta$$:

$$
\cos^2 	heta = \frac{1 + \cos 2	heta}{2}
$$  
This will simplify the integration process.

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.

The region inside the limaçon r = 2 + cos θ

Verified video answer for a similar problem:

Key Concepts

Polar Coordinates and Graphing

Area Calculation in Polar Coordinates

Identifying Limits of Integration