Explanation Given: A parallelogram Pin the xy-plane with vertices ( 0 , 0 ) , ( 2 , 2 ) , ( 1 , 4 ) , ( 3 , 6 ) and a rectangle [ 0 , 1 ] × [ 0 , 1 ] . Formula Used: 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 .... ( 1 ) Jac ( G ) = | ∂ x ∂ u ∂ x ∂ v ∂ y ∂ u ∂ y ∂ v | ... ( 2 ) Calculation: Assume the linear map is: T ( u , v ) = ( A u + B v , C u + D v ) The vertices of the given rectangle [ 0 , 1 ] × [ 0 , 1 ] are: ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) Here, the mapping is linear. So, T ( 0 , 0 ) = ( 0 , 0 ) T ( 0 , 1 ) = ( 1 , 4 ) T ( 1 , 0 ) = ( 2 , 2 ) T ( 1 , 1 ) = ( 3 , 6 ) Now, T ( 0 , 1 ) = ( 1 , 4 ) ( 0 + B , 0 + D ) = ( 1 , 4 ) B = 1 , D = 4 And T ( 1 , 0 ) = ( 2 , 2 ) ( A + 0 , C + 0 ) = ( 2 , 2 ) A = 2 , C = 2 Thus, the required linear mapping is: T ( u , v ) = ( 2 u + v , 2 u + 4 v ) Now, use the formula (2) to calculate the double integral of e 2 x − y over P and we get: ∬ P e 2 x − y d A = ∫ 0 1 ∫ 0 1 e 2 ( 2 u + v ) − ( 2 u + 4 v ) | ∂ ( x , y )

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ISBN: 9781319434526

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.6, Problem 24E

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A linear map T that maps $[0, 1] \times [0, 1]$ to the parallelogram Pin the xy-plane with vertices $(0, 0), (2, 2), (1, 4), (3, 6)$ and also, the double integral of $e^{2 x - y}$ over P via change of variables.

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Chapter 16.6, Problem 23E

Chapter 16.6, Problem 25E

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Chapter 16 Solutions

CALCULUS 4E (LL) W/ ACHIEVE ACCESS

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A linear map T that maps [ 0 , 1 ] × [ 0 , 1 ] to the parallelogram Pin the xy-plane with vertices ( 0 , 0 ) , ( 2 , 2 ) , ( 1 , 4 ) , ( 3 , 6 ) and also, the double integral of e 2 x − y over P via change of variables.

Solution Summary: The author calculates a linear map T that maps to the parallelogram Pin the xy-plane with vertices (0, 0) and the double integral of e 2 y over P

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To calculate:

A linear map T that maps $[0, 1] \times [0, 1]$ to the parallelogram Pin the xy-plane with vertices $(0, 0), (2, 2), (1, 4), (3, 6)$ and also, the double integral of $e^{2 x - y}$ over P via change of variables.

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Chapter 16 Solutions

A linear map T that maps [ 0 , 1 ] × [ 0 , 1 ] to the parallelogram Pin the xy-plane with vertices ( 0 , 0 ) , ( 2 , 2 ) , ( 1 , 4 ) , ( 3 , 6 ) and also, the double integral of e 2 x − y over P via change of variables.

Solution Summary: The author calculates a linear map T that maps to the parallelogram Pin the xy-plane with vertices (0, 0) and the double integral of e 2 y over P

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To calculate: A linear map T that maps [0,1]×[0,1]to the parallelogram Pin the xy-plane with vertices (0,0),(2,2),(1,4),(3,6)and also, the double integral of e2x−yover P via change of variables.

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Chapter 16 Solutions

To calculate:

A linear map T that maps $[0, 1] \times [0, 1]$ to the parallelogram Pin the xy-plane with vertices $(0, 0), (2, 2), (1, 4), (3, 6)$ and also, the double integral of $e^{2 x - y}$ over P via change of variables.