Explanation Given: The double integral is, ∬ R x 1 + x 2 y 2 d A .Also the indicated change in variable is, x = u ; y = v u . Formula used: Jacobian ∂ ( x , y ) ∂ ( u , v ) = ( ∂ x ∂ u ⋅ ∂ y ∂ v ) − ( ∂ x ∂ v ⋅ ∂ y ∂ u ) To find the area using change of variable, ∬ R f ( x , y ) d x d y = ∬ S f ( g ( u , v ) , h ( u , v ) ) | ∂ ( x , y ) ∂ ( u , v ) | d u d v Integration formula: ∫ 1 1 + v 2 d v = arctan ( v 1 ) + c Calculation: Consider the provided figure, As provided in this problem x = u and y = v u . Find the Jacobian ∂ ( x , y ) ∂ ( u , v ) , ∂ ( x , y ) ∂ ( u , v ) = | ∂ x ∂ u ∂ x ∂ v ∂ y ∂ u ∂ y ∂ v | = | 1 0 v ln u 1 u | = 1 ( 1 u ) − 0 = 1 u Solve provided equation to get the value of u and v . x = u ; y = v u Solve the above two equation, to get u = x and v = x y Rewrite the equation of provided four line by which region R is bounded, x = 1 ; x = 5 ; x y = 1 ; x y = 5 From above four equations it can be understood that 1 ≤ x ≤ 5 that is 1 ≤ u ≤ 5 and 1 ≤ x y ≤ 5 that is, 1 ≤ v ≤ 5 . Thus, bounds on u and v are 1 ≤ u ≤ 5 and 1 ≤ v ≤ 5 respectively

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Chapter 14, Problem 78RE

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To calculate: The value of provided double integral ∬Rx1+x2y2dA by the use of indicated change in variables x=u;y=vu.

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Chapter 14, Problem 77RE

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The value of provided double integral ∬ R x 1 + x 2 y 2 d A by the use of indicated change in variables x = u ; y = v u .

Solution Summary: The author explains how to calculate the value of provided double integral by the use of indicated change in variables x=u;y=vu

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To calculate: The value of provided double integral ∬Rx1+x2y2dA by the use of indicated change in variables x=u;y=vu.

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Chapter 14 Solutions