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 = 2 + sin θ and r = 3 sin θ . Use online graphing calculator and draw the graph of r = 2 + sin θ and r = 3 sin θ as shown below in Figure 1. In Figure 1, the shaded region is the required region. The area of the required region A is obtained by subtracting the unshaded part of circle from the area of circle. Let the area of the circle be A 1 . The area A 1 is bounded between r = 2 + sin θ and r = 3 sin θ . Substitute r = 2 + sin θ in r = 3 sin θ to find the intersection point of the curve. 2 + sin θ = 3 sin θ 2 sin θ = 2 sin θ = 1 2 θ = π 4 or θ = 3 π 4 From Figure 1 it is noted that the circle is at a height of 3 units so the radius is 3 2 . Therefore, the area A 1 is, A 1 = π ( 3 2 ) 2 = 9 π 4 Therefore, the area is A 1 = 9 π 4 . The area of the unshaded part of the circle be A 2 . The area A 2 is obtained by subtracting the area of the curve r = 2 + sin θ inside the circle. Evaluate the area A 2 as follows. A 2 = ∫ π 4 π 2 1 2 ( ( 3 sin θ ) 2 − ( 2 + sin θ ) 2 ) d θ = 1 2 ∫ π 4 π 2 ( 9 sin 2 θ − ( 2 + sin 2 θ + 2 2 sin θ ) ) d θ = 1 2 ∫ π 4 π 2 ( 9 sin 2 θ − 2 − sin 2 θ − 2 2 sin θ ) d θ = 1 2 ∫ π 4 π 2 ( 8 sin 2 θ − 2 − 2 2 sin θ ) d θ Further simplify as follows

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

3rd Edition

ISBN: 9780134996714

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

Publisher: PEARSON

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Chapter 12, Problem 48RE

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The area of the region that is inside r=2+sinθ and inside r=3sinθ.

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Chapter 12, Problem 47RE

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

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The area of the region that is inside r = 2 + sin θ and inside r = 3 sin θ .

Solution Summary: The area inside the polar curve r=f(theta ) is given by A=displaystyle

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The area of the region that is inside r=2+sinθ and inside r=3sinθ.

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