Question

45–60. Areas of regions Find the area of the following regions.The region common to the circles r = 2 sin θ and r = 1

Accepted Answer

First, understand the problem: we need to find the area of the region common to the two circles given in polar coordinates: $$r = 2 \sin 	heta$$ and $$r = 1. $$Step 1: Sketch or visualize the two circles to understand their intersection points and the region they enclose. The circle $$r = 2 \sin 	heta is a $$circle centered at $$(0,1)$$ with radius 1, and $$r = 1 is a $$circle centered at the origin with radius 1. Step 2: Find the points of intersection by setting the two equations equal: $$2 \sin 	heta = 1. $$Solve for $$	heta to $$find the limits of integration for the common region. Step 3: Set up the integral for the area of the common region. The area enclosed by a polar curve $$r(	heta)$$ from $$\alpha to$$ $$\beta is $$given by $$\frac{1}{2} \int_{\alpha}^{\beta} r(\$$theta)^2$$ \, d	heta. $$For the common region, the area will be the sum of areas bounded by each curve between the intersection angles, taking care to integrate the smaller radius where appropriate. Step 4: Write the integral expressions for the area of the common region by splitting the integral at the intersection points and using the appropriate $$r(	heta)$$ for each part. Then, prepare to evaluate these integrals to find the total area.

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

The region common to the circles r = 2 sin θ and r = 1

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Concetti chiave

Polar Coordinates and Graphing

Area Calculation in Polar Coordinates

Finding Intersection Points of Polar Curves