Explanation The given curve is r = 1 + 2 sin θ . Use online graphing calculator to sketch the graph of the curve r = 1 + 2 sin θ as shown below in Figure 1. From the above figure, it can be observed that the graph is symmetric about the y- axis. Notice that, one half of the smaller loop starts for θ = 7 π 6 to θ = 3 π 2 . So, the area inside the loop can be evaluated as follows. Area = 2 × 1 2 ∫ 7 π 6 3 π 2 ( 1 + 2 sin θ ) 2 d θ = ∫ 7 π 6 3 π 2 ( 1 + 4 sin θ + 4 sin 2 θ ) d θ = ∫ 7 π 6 3 π 2 ( 1 + 4 sin θ + 2 − 2 cos 2 θ ) d θ = ∫ 7 π 6 3 π

3rd Edition

ISBN: 9780135873311

Author: Briggs

Publisher: PEARSON CO

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Chapter 12, Problem 45RE

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To find: The area of the region inside the loop r=1+2sinθ.

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Chapter 12, Problem 44RE

Chapter 12, Problem 46RE

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The area of the region inside the loop r = 1 + 2 sin θ .

Solution Summary: The author illustrates how the graph of the curve r=1+2mathrmsintheta is symmetric about the y- axis.