Explanation Given: The integral is ∬ R x 2 + y 2 d A . The region is R = { ( x , y ) : 1 ≤ x 2 + y 2 ≤ 4 } . Formula used: If f is a polar rectangle R given by 0 ≤ a ≤ r ≤ b , α ≤ θ ≤ β , where 0 ≤ β − α ≤ 2 π , then, ∬ R f ( x , y ) d A = ∫ α β ∫ a b f ( r cos θ , r sin θ ) r d r d θ (1) If g ( x ) is the function of x and h ( y ) is the function of y then, ∫ a b ∫ c d g ( x ) h ( y ) d y d x = ∫ a b g ( x ) d x ∫ c d h ( y ) d y (2) Calculation: From the given condition, it is observed that the given region is the annulus which contains two circles of radii 1 and 4. Therefore, r varies from 1 to 4 and θ varies from 0 to 2 π . Substitute x = r cos θ , y = r sin θ to convert the problem into polar coordinates, because it is easy to evaluate integral with polar coordinates when equation of circle is given

Calculus: Early Transcendentals, Books a la Carte Plus MyLab Math/MyLab Statistics Student Access Kit (2nd Edition)

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

Publisher: PEARSON

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Chapter 13.3, Problem 55E

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Chapter 13.3, Problem 54E

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Chapter 13 Solutions

Calculus: Early Transcendentals, Books a la Carte Plus MyLab Math/MyLab Statistics Student Access Kit (2nd Edition)

Ch. 13.1 - 1. Write an iterated integral that gives the...Ch. 13.1 - Write an iterated integral that gives the volume...Ch. 13.1 - Write two iterated integrals that equal Rf(x,y)dA,...Ch. 13.1 - Consider the integral 1311(2y2+xy)dydx. State the...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...

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