Chapter 14.3, Problem 48E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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+2cosÎ¸ 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

Î¸=0,Ï€6,Ï€4,Ï€3,Ï€2

 Î¸ r1=1 r2=2+2cosÎ¸ (r1,Î¸),(r2,Î¸) 0 1 4 (1,0),(4,0) Ï€6 1 2+3 (1,Ï€6),(2+3,Ï€6) Ï€4 1 2+2 (1,Ï€4),(2+2,Ï€4) Ï€3 1 3 (1,Ï€3),(3,Ï€3) Ï€2 1 2 (1,Ï€2),(2,Ï€2)

Mark the points on the plot and connect the points.

Use the symmetry to complete the graph for the given limit, that is, Î¸=2Ï€.

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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