Explanation Given: Let G ( u , v ) = ( u v + 1 , u v v + 1 ) . Formula Used: Jac ( G ) = | ∂ x ∂ u ∂ x ∂ v ∂ y ∂ u ∂ y ∂ v | ... ( 1 ) If G : D 0 → D and if f is continuous then, ∬ D f ( x , y ) d x d y = ∬ D 0 f ( x ( u , v ) , y ( u , v ) ) | ∂ ( x , y ) ∂ ( u , v ) | d u d v .... ( 2 ) Calculation: Given mapping is: G ( u , v ) = ( u v + 1 , u v v + 1 ) The given region in xy-plane is bounded by: y = x , y = 2 x , x + y = 3 and x + y = 6 We have to find ∬ D ( x + y ) d x d y . So, the transformations will be: x = u v + 1 , y = u v v + 1 → y x = v y = u v v + 1 → y = u − u v + 1 → y = u − x → x + y = u 1 ≤ y x ≤ 2 , 3 ≤ x + y ≤ 6 → 1 ≤ v ≤ 2 , 3 ≤ u ≤ 6 To calculate the area of G ( R ) , first we need to find the Jacobian by using the formula(1): x = u v + 1 ∂ x ∂ u = 1 v + 1 , ∂ x ∂ v = − u ( v + 1 ) 2 And y = u − u v + 1 ∂ y ∂ u = 1 − 1 v + 1 , ∂ y ∂ v = u ( v + 1

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ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 16.6, Problem 32E

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To calculate:

$\iint_{D} (x + y) d x d y$ ,where D is the shaded region in the given figure.

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Chapter 16.6, Problem 31E

Chapter 16.6, Problem 33E

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Chapter 16 Solutions

CALCULUS W/SAPLING ACCESS >IC<

Ch. 16.1 - Prob. 1PQ Ch. 16.1 - Prob. 2PQ Ch. 16.1 - Prob. 3PQ Ch. 16.1 - Prob. 4PQ Ch. 16.1 - Prob. 5PQ Ch. 16.1 - Prob. 6PQ Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E

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∬ D ( x + y ) d x d y ,where D is the shaded region in the given figure.

Solution Summary: The given region in xy-plane is bounded by: y = u v + 1 and if f is continuous then the transformations will be:

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To calculate:

$\iint_{D} (x + y) d x d y$ ,where D is the shaded region in the given figure.

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Chapter 16 Solutions

∬ D ( x + y ) d x d y ,where D is the shaded region in the given figure.

Solution Summary: The given region in xy-plane is bounded by: y = u v + 1 and if f is continuous then the transformations will be:

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To calculate: ∬D(x+y)dxdy,where D is the shaded region in the given figure.

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Chapter 16 Solutions

To calculate:

$\iint_{D} (x + y) d x d y$ ,where D is the shaded region in the given figure.