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: The given equations of the polar curves are r = cos θ and r = 1 2 . Use online graphing calculator and draw the graph of r = cos θ and r = 1 2 as shown below in Figure 1. In Figure 1, the shaded region is the required region. The area outside the circle θ = 1 2 and inside the curve r = cos θ is symmetric about positive side of the x- axis. The half of this region is generated when θ varies from 0 to π 3 . Therefore, the area of the half of the region A 1 is, A 1 = ∫ 0 π 3 1 2 ( ( cos θ ) 2 − ( 1 2 ) 2 ) d θ = 1 2 ∫ 0 π 3 cos θ d θ

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Chapter 12.3, Problem 46E

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To find: The area of the region that is outside the circle r=12 and inside the curve r=cosθ with the help of the sketch of the region and its bounding curves.

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Chapter 12.3, Problem 45E

Chapter 12.3, Problem 47E

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Chapter 12 Solutions

Pearson eText Calculus: Early Transcendentals -- Instant Access (Pearson+)

Ch. 12.1 - Identify the graph generated by the parametric...Ch. 12.1 - Prob. 2QC Ch. 12.1 - Prob. 3QC Ch. 12.1 - Prob. 4QC Ch. 12.1 - Prob. 5QC Ch. 12.1 - Prob. 6QC Ch. 12.1 - Explain how a pair of parametric equations...Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E

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The area of the region that is outside the circle r = 1 2 and inside the curve r = cos θ with the help of the sketch of the region and its bounding curves.

Solution Summary: The author calculates the area of the region outside the circle r=1sqrt2 and inside the curve with the help of a sketch and its bounding curves.

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To find: The area of the region that is outside the circle r=12 and inside the curve r=cosθ with the help of the sketch of the region and its bounding curves.

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Chapter 12 Solutions