16.5 Ex 21 - 22

# 16.5 Ex 21 - 22 - 1000 C H A P T E R 16 M U LTI P L E I N T...

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1000 CHAPTER 16 MULTIPLE INTEGRATION (ET CHAPTER 15) 8( 1 , 0 ) = ( A · 1 + C · 0 , B · 1 + D · 0 ) = ( A , B ) = ( 1 , 7 ) A = 1 , B = 7 We substitute in (1) to obtain the following map: u ,v) = ( u 2 v, 7 u + 5 v) 21. Let D be the parallelogram in Figure 14. Apply the Change of Variables Formula to the u = ( 5 u + 3 u + 4 to evaluate ZZ D xydA as an integral over D 0 =[ 0 , 1 ]×[ 0 , 1 ] . y x D (5, 1) (3, 4) FIGURE 14 SOLUTION v u D D 0 y x Φ 1 (5, 1) 1 (3, 4) We express f ( x , y ) = xy in terms of u and v .Since x = 5 u + 3 v and y = u + 4 v ,wehave f ( x , y ) = = ( 5 u + 3 v)( u + 4 = 5 u 2 + 12 v 2 + 23 u v The Jacobian of the linear map u = ( 5 u + 3 u + 4 is Jac (8) = ∂( x , y ) u = ¯ ¯ ¯ ¯ 53 14 ¯ ¯ ¯ ¯ = 20 3 = 17 Applying the Change of Variables Formula we get D = D 0 f ( x , y ) ¯ ¯ ¯ ¯ x , y ) u ¯ ¯ ¯ ¯ dud v = Z 1 0 Z 1 0 ( 5 u 2 + 12 v 2 + 23 u · 17 v = 17 Z 1 0 5 u 3 3 + 12 v 2 u + 23 u 2 v 2 ¯ ¯ ¯ ¯ 1 u = 0 d v = 17 Z 1 0 µ 5 3 + 12 v 2 + 23 v 2 d v = 17 ± 5 v 3 + 4 v 3 + 23 v 2 4 ¯ ¯ ¯ ¯ 1 0 ! = 17 µ 5 3 + 4 + 23 4 = 2329 12 194 . 08 22. Let u = ( u u u of the following sets: (a) Determine the images of the horizontal and vertical lines in the u v -plane.

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16.5 Ex 21 - 22 - 1000 C H A P T E R 16 M U LTI P L E I N T...

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