   Chapter 14.3, Problem 48E

Chapter
Section
Textbook Problem

Area:In Exercises 47–52, sketch a graph of the region bounded by the graphs of the equations. Then use a double integral to find the area of the region.Inside the cardioid r = 2 + 2 cos θ and outside the circle r = 1 Area:In Exercises 47–52, sketch a graph of the region bounded by the graphs of the equations. Then use a double integral to find the area of the region.Inside the cardioid r = 2 + 2 cos θ and outside the circle r = 1

To determine

To calculate: The area of the shaded region by plotting a graph using the equation r=2+2cosθ and r=1.

Explanation

Given:

The values of r are r=2+2cosθ and r=1.

Calculation:

From the given equations, draw the graph. Since we have two values for the radius, we first draw the two circles with the given equations by performing the following steps:

Draw the reference axis in polar coordinates.

Find the values of r for

θ=0,π6,π4,π3,π2

 θ r1=1 r2=2+2cosθ (r1,θ),(r2,θ) 0 1 4 (1,0),(4,0) π6 1 2+3 (1,π6),(2+3,π6) π4 1 2+2 (1,π4),(2+2,π4) π3 1 3 (1,π3),(3,π3) π2 1 2 (1,π2),(2,π2)

Mark the points on the plot and connect the points.

Use the symmetry to complete the graph for the given limit, that is, θ=2π.

The area of integration lies within the limits of the integral, so shade according to the given conditions.

The final plot is shown below:

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