Question

45–60. Areas of regions Find the area of the following regions.The region outside the circle r = 1/2 and inside the circle r = cos θ

Accepted Answer

First, understand the problem: we want to find the area of the region that lies outside the circle given by $$r = \frac{1}{2}$$ and inside the circle given by $$r = \cos 	heta in $$polar coordinates. Next, find the points of intersection between the two circles by setting their equations equal: $$\frac{1}{2} = \cos 	heta. $$Solve this equation for $$	heta to $$determine the limits of integration. Recall that the area enclosed by a polar curve $$r = f(	heta)$$ between angles $$\alpha$$ and $$\beta is $$given by the integral $$\frac{1}{2} \int_{\alpha}^{\beta} (f(\$$theta))^2$$ \, d	heta. We $$will use this formula to find the areas inside each circle. Set up the integral for the area inside $$r = \cos 	heta$$ but outside $$r = \frac{1}{2} by $$subtracting the area inside $$r = \frac{1}{2}$$ from the area inside $$r = \cos 	heta$$ over the interval between the intersection points found in step 2. Finally, evaluate the integral expressions (or leave them in integral form if not asked to compute) to express the area of the desired region.

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

The region outside the circle r = 1/2 and inside the circle r = cos θ

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Key Concepts

Polar Coordinates and Graphs

Area Calculation in Polar Coordinates

Determining Intersection Points and Limits of Integration