Explanation Given: The double integral ∫ 0 1 ∫ x 3 1 ( x + y 2 ) d y d x Concept Used: A double integral ∬ R f ( x , y ) d A over the range R : a ≤ x ≤ b , g 1 ( x ) ≤ y ≤ g 2 ( x ) is defined as ∬ R f ( x , y ) d A = ∫ a b ∫ g 1 ( x ) g 2 ( x ) f ( x , y ) d y d x And A double integral ∬ R f ( x , y ) d A over the range R : m 1 ( x ) ≤ x ≤ m 2 ( x ) , c ≤ y ≤ d is defined as ∬ R f ( x , y ) d A = ∫ c d ∫ m 1 ( x ) m 2 ( x ) f ( x , y ) d x d y Calculation: For the givenintegral ∫ 0 1 ∫ x 3 1 ( x + y 2 ) d y d x With the range of rectangle R : 0 ≤ x ≤ 1 , x 3 ≤ y ≤ 1 . Sketching the region of integration Changing the order of integration ∫ 0 1 ∫ x 3 1 ( x + y 2 ) d y d x = ∫ y = 0 1 ∫ x = 0 y 1 3 ( x + y 2 ) d x d y Evaluating the integral ∫ 0 1 ∫ x 3 1 ( x + y 2 ) d y d x = ∫ 0 1 [ x y + y 3 3 ] y = <

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ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 12, Problem 4SP

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Tosketch the region of integration ∫01∫ x 31( x+ y 2 )dydx , exchange the order and evaluate the integral.

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Chapter 12, Problem 3SP

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Chapter 12 Solutions

Calculus

Ch. 12.1 - Prob. 1PS Ch. 12.1 - Prob. 2PS Ch. 12.1 - Prob. 3PS Ch. 12.1 - Prob. 4PS Ch. 12.1 - Prob. 5PS Ch. 12.1 - Prob. 6PS Ch. 12.1 - Prob. 7PS Ch. 12.1 - Prob. 8PS Ch. 12.1 - Prob. 9PS Ch. 12.1 - Prob. 10PS

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Tosketch the region of integration ∫ 0 1 ∫ x 3 1 ( x + y 2 ) d y d x , exchange the order and evaluate the integral.

Solution Summary: The author analyzes the region of integration and evaluates it.

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Tosketch the region of integration ∫01∫ x 31( x+ y 2 )dydx , exchange the order and evaluate the integral.

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Chapter 12 Solutions