Explanation Given: The double integral: ∬ D e x y d x d y Where region D is graphed below: Formula used: Jacobian = [ ∂ x ∂ u ∂ x ∂ v ∂ y ∂ u ∂ y ∂ v ] Calculation: To evaluate the given double integral, we must first choose an appropriate set of equations that map the given region appropriately to a well-known shape. Thus, we can choose the map as: u = x y , v = x 2 y ⇒ x = v u , y = u 2 v Plugging these substitutions in the given four equations, we get our bounds for the new integral as 10 ≤ u ≤ 20 , 20 ≤ v ≤ 40 The new region looks like this: Let us now evaluate the Jacobian for this map: Jacobian = [ ∂ x ∂ u ∂ x ∂ v ∂ y ∂ u ∂ y ∂ v ] = [ ∂ ∂ u ( v u ) ∂ ∂ v ( v u ) ∂ ∂ u ( u 2 v ) ∂ ∂ v ( u 2 v ) ] Jacobian = [ − v u 2 1 u 2 u v − u 2 v 2 ] = ( v u 2 ) ( u 2 v 2 ) − ( 1 u ) ( 2 u v ) Jacobian = 1 v − 2 v = − 1 v Thus, our integral becomes: ⇒ ∬ D e x y d x d y = ∫ 10 20 ∫ 20 40 e ( v u ) ( u 2 v ) | J | d v d u

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 16.6, Problem 39E

To determine

To prove:

That the double integral converts as per the following formula when change of variables is applied.

$\iint_{D} e^{x y} d x d y = \int_{10}^{20} \int_{20}^{40} e^{u} v^{- 1} d v d u$

To evaluate:

The integral.

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Chapter 16.6, Problem 38E

Chapter 16.6, Problem 40E

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Chapter 16 Solutions

Calculus - Standalone book

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To prove: That the double integral converts as per the following formula when change of variables is applied. ∬ D e x y d x d y = ∫ 10 20 ∫ 20 40 e u v − 1 d v d u To evaluate: The integral.

Solution Summary: The author proves that the double integral converts as per the following formula when change of variables is applied.

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To prove:

That the double integral converts as per the following formula when change of variables is applied.

$\iint_{D} e^{x y} d x d y = \int_{10}^{20} \int_{20}^{40} e^{u} v^{- 1} d v d u$

To evaluate:

The integral.

Want to see the full answer?

Chapter 16 Solutions

To prove: That the double integral converts as per the following formula when change of variables is applied. ∬ D e x y d x d y = ∫ 10 20 ∫ 20 40 e u v − 1 d v d u To evaluate: The integral.

Solution Summary: The author proves that the double integral converts as per the following formula when change of variables is applied.

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To prove: That the double integral converts as per the following formula when change of variables is applied. ∬Dexydxdy=∫1020∫2040euv−1dvdu To evaluate: The integral.

Want to see the full answer?

Chapter 16 Solutions

To prove:

That the double integral converts as per the following formula when change of variables is applied.

$\iint_{D} e^{x y} d x d y = \int_{10}^{20} \int_{20}^{40} e^{u} v^{- 1} d v d u$

To evaluate:

The integral.