Explanation Given: The region R is the square with vertices ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 0 ) and ( 1 , − 1 ) . Property used: Change of Variable Change of Variable in double integral is given by, ∬ R f ( x , y ) d A = ∬ S f ( x ( u , v ) , y ( u , v ) ) | ∂ ( x , y ) ∂ ( u , v ) | d u d v (1) Calculation: Use the transformation u = x − y , v = x + y . First find the value of x and y. Add the two transformations will yield, 2 x = u + v . Thus, x = 1 2 ( u + v ) . Similarly, subtract two transformations will give − 2 y = u − v . Thus, y = 1 2 ( v − u ) . Obtain the Jacobian, ∂ ( x , y ) ∂ ( u , v ) = | ∂ x ∂ u ∂ x ∂ v ∂ y ∂ u ∂ y ∂ v | Find the partial derivative of x and y with respect to u and v. x = 1 2 ( u + v ) then x u = 1 2 and x v = 1 2 and y = 1 2 ( v − u ) then y u = − 1 2 and y v = 1 2 . ∂ ( x , y ) ∂ ( u , v ) = | 1 2 1 2 − 1 2 1 2 | = 1 2 ( 1 2 ) − ( 1 2 ) ( − 1 2 ) = 1 4 + 1 4 = 1 2 From the given conditions, rewrite the given integral as given below. x y = 1 2 ( u + v ) 1 2 ( u − v ) = 1 4 ( u 2 − v 2 ) Find the boundary by using the given transformation. For the point ( 0 , 0 ) , u = 0 , v = 0 . For the point ( 1 , 1 ) , u = − 2 , v = 0 . For the point ( 2 , 0 ) , u = − 2 , v = 2

Calculus: Early Transcendentals, Loose-leaf Version, 9th

9th Edition

ISBN: 9780357022290

Author: Stewart

Publisher: Cengage Learning Acquisitions

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Chapter 15, Problem 57E

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To evaluate: The integral by change of variable method.

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Chapter 15, Problem 56E

Chapter 15, Problem 58E

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Chapter 15 Solutions

Calculus: Early Transcendentals, Loose-leaf Version, 9th

Ch. 15.1 - (a) Estimate the volume of the solid that lies...Ch. 15.1 - If R = [0, 4] [1, 2], use a Riemann sum with m =...Ch. 15.1 - (a) Use a Riemann sum with m = n = 2 to estimate...Ch. 15.1 - (a) Estimate the volume of the solid that lies...Ch. 15.1 - Let V be the volume of the solid that lies under...Ch. 15.1 - A 20-ft-by-30-ft swimming pool is filled with...Ch. 15.1 - A contour map is shown for a function f on the...Ch. 15.1 - Evaluate the double integral by first identifying...Ch. 15.1 - Evaluate the double integral by first identifying...Ch. 15.1 - Evaluate the double integral by first identifying...

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The integral by change of variable method.

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Chapter 15 Solutions