Explanation Result used: The area inside the polar curve r = f ( θ ) over the interval [ α , β ] is given by A = ∫ α β 1 2 f ( θ ) 2 d θ . Calculation: Equate the given curves to obtain the intersection points as follows. 4 + cos 2 θ = 3 cos 2 θ 4 = 2 cos 2 θ cos 2 θ = 2 The equation cos 2 θ = 2 will never yiled to a solution. Thus, use th egraph to obtain the intersection points. Use online graphing calculator and draw the curves r = 4 + cos 2 θ and r = 3 cos 2 θ and identify the intersection points as shown below in Figure 1. From Figure 1, it can be observed that the curves intersect at the points ( 3 , π 2 ) and ( 3 , − π 2 ) . Hence, the intersection points of the curves r = 4 − 2 cos θ and r = 2 + 2 cos θ are ( 3 , π 2 ) and ( 3 , − π 2 ) . The area of the region in first quadrant inside the curve r = 4 + cos 2 θ but outside the curve r = 3 cos 2 θ can be evaluated as follows. The required area is generated when θ varies from 0 to π 2 for the curve r = 4 + cos 2 θ and from 0 to π 4 for the curve r = 3 cos 2 θ . Notice that, there are two half leaves of the curve r = 3 cos 2 θ inside the curve r = 4 + cos 2 θ in the first quadrant. The required area is area inside the curve r = 4 + cos 2 θ minus area under the curve r = 3 cos 2 θ for two half leaves. A r e a = 1 2 ∫ 0 π 2 ( 4 + cos 2 θ ) 2 d θ − 2 × 1 2 ∫ 0 π 4 ( 3 cos 2 θ ) 2 d θ = 1 2 ∫ 0 π 2 ( 16 + 8 cos 2 θ +

Calculus: Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)

3rd Edition

ISBN: 9780134995991

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 12.3, Problem 62E

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To find: All the intersection points for the curves r=4+cos2θ and r=3cos2θ and also find the areas of the region in first quadrant inside the curve r=4+cos2θ but outside the curve r=3cos2θ.

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Chapter 12.3, Problem 61E

Chapter 12.3, Problem 63E

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Chapter 12 Solutions

Calculus: Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)

Ch. 12.1 - Identify the graph generated by the parametric...Ch. 12.1 - Prob. 2QC Ch. 12.1 - Describe the curve generated by x = 3 + 2t, y = 12...Ch. 12.1 - Find parametric equations for the line segment...Ch. 12.1 - Use Theorem 12.1 to find the slope of the line x =...Ch. 12.1 - Use the arc length formula to find the length of...Ch. 12.1 - Explain how a pair of parametric equations...Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Give parametric equations that generate the line...

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All the intersection points for the curves r = 4 + cos 2 θ and r = 3 cos 2 θ and also find the areas of the region in first quadrant inside the curve r = 4 + cos 2 θ but outside the curve r = 3 cos 2 θ .

Solution Summary: The author calculates the intersection points of the curves using a graph.