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. Observe that the shaded region is consisting of two subregions. One region say R 1 is bounded the circumference of the circle r = 1 2 and other say R 2 is bounded by the outer line of the curve r = cos θ . Therefore, the area of the required region is the sum of the areas of these two regions. From Figure 1 it can be noted that the line θ = π 3 divides the two regions. Area of the region R 1 is given by A 1 = ∫ 0 π 3 1 2 ( 1 2 ) 2 d θ and area of the region R 2 is given by A 2 = ∫ π 3 π 2 1 2 ( cos θ ) d θ

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

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

Publisher: PEARSON

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Chapter 12.3, Problem 47E

To determine

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

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Chapter 12.3, Problem 46E

Chapter 12.3, Problem 48E

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Chapter 12 Solutions

Calculus, Single Variable: Early Transcendentals (3rd Edition)

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 inside the circle r = 1 2 and inside the curve r = cos θ in the first quadrant with the help of the sketch of the region and its bounding curves.

Solution Summary: The author calculates the area of the region inside the circle r=1sqrt2 and within the curve using an online graphing calculator.

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

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Chapter 12 Solutions