Explanation Description: The region of 0 ≤ r 1 ( θ ) ≤ r ≤ r 2 ( θ ) lies between the two polar curves r 1 = r 1 ( θ ) and r 2 = r 2 ( θ ) from θ = α to θ = β . Therefore, the area of the region of 0 ≤ r 1 ( θ ) ≤ r ≤ r 2 ( θ ) is the difference between the areas of r 2 = r 2 ( θ ) and r 1 = r 1 ( θ ) . From the analysis, it is clear that, the required area of the region is calculated as follows: Find the areas of individual curves from θ = α to θ = β . Subtract the area of the curve r 1 = r 1 ( θ ) from the area of the curve r 2 = r 2 ( θ ) in order to obtain the area of required region. Consider the formula for area of the region between the origin and the curve. A = ∫ α β 1 2 r 2 d θ From the procedure, modify the formula to find the area of the required region as follows: A = ∫ α β 1 2 ( r 2 ) 2 d θ − ∫ α β 1 2 ( r 1 ) 2 d θ Simplify the expression as follows: A = 1 2 ∫ α β [ ( r 2 ) 2 − ( r 1 ) 2 ] d θ (1) Example: Consider a region 0 ≤ 1 ≤ r ≤ 2 sin θ , π 6 ≤ θ ≤ 5 π 6 . The area of this region can be calculated as follows: From the given region write the required parameters as follows: r 1 = 1 r 2 = 2 sin θ α = π 6 β = 5 π 6 Substitute 1 for r 1 , 2 sin θ for r 2 , π 6 for α , and 5 π 6 for β in Equation (1) to obtain the area of the required region as follows: A = 1 2 ∫ π 6 5 π 6 [ ( 1 ) 2 − ( 2 sin θ ) 2 ] d θ = ∫ π

Thomas' Calculus - MyMathLab Integrated Review

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Chapter 11, Problem 12GYR

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Describe the procedure to find the area of a region 0≤r1(θ)≤r≤r2(θ), α≤θ≤β, in the polar coordinate plane and provide the examples.

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Ch. 11.1 - Finding Cartesian from Parametric...Ch. 11.1 - Finding Cartesian from Parametric...Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - Prob. 8E Ch. 11.1 - Prob. 9E Ch. 11.1 - Finding Cartesian from Parametric...

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Describe the procedure to find the area of a region 0 ≤ r 1 ( θ ) ≤ r ≤ r 2 ( θ ) , α ≤ θ ≤ β , in the polar coordinate plane and provide the examples.

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Describe the procedure to find the area of a region 0≤r1(θ)≤r≤r2(θ), α≤θ≤β, in the polar coordinate plane and provide the examples.

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Chapter 11 Solutions