Question

To evaluate the following integrals carry out these steps. a. Sketch the original region of integration R in the xy-plane and the new region S in the uv-plane using the given change of variables. b. Find the limits of integration for the new integral with respect to u and v. c. Compute the Jacobian. d. Change variables and evaluate the new integral. х x,y): 0 5x57, 7 sys 6 - -x}; use x=7u, y = 6v - u. S5x25x+7y da, where R= -x} R a. Sketch the original region of integration R in the xy-plane. Choose the correct graph below. A. B. 0 Ау лу 2- AY 2- Ay 8 8 Sketch the new region S in the uv-plane. Choose the correct answer below. O A. B. OC. V v Av 2- 3- 1 2- Ź 2 N- 21

Answer 1

พ” R To find SS x+zy dA OSX 27, -2 Lysbn) (a). The region Ŕ shown as the figure Where R = {(1,4): O :y in 24 plane : Y 1810,6) x+y=6 x=7 (7,0) 8 A 67,-1), + yaan 7 To find the point of intersection A. from two lines x+y=6, and x=7, to get y = -1 so, point of Ais (7,4) Therefore option (D) is correct. Consider the transformation - *770, y = 6ie-u, Then the region R transform to the region sin s. in un plane. osuz.7 transform itd oluli y = - 2/3 tranx form to 6.- U= - 14. (74) or, 20 The line so, 2 y = 6x transform to 60-u = 6-74 or, D +U=1, *** .

Thus the Yenne - + 1 4 6-х vossform to o219 2 1-ц, regions The new in un plane is S= { tv, 9): 02 ч 21, 2 # 1- The new region s shown figure - -че in the У 1) va 9. 1-4 4,0) A >u 2 The new ! regions sin un plane is option (B) by we can write the result of new regions- о) 2 и 41 o219 2 1- и ЪТ 7 Т(u,9) . КQ he ne Ох 19 he 29 7 - 1 | [н: 24 д - 60 - 7 о 42 J(ч,9) “ А2 .

), S x Jx+zy dA R - 카u+키 (60-u) 15{u,에 dudd S I-u 므 49년 JJu+420-Fu . 42 dudly Uc6 1920 리 내 5 dudd. Use 1920 u = 45 423 JS -3> 를 J [4] 을 (4 Joe (trian - 49 42) du U스 1920 11. putting 1-u au 아, du = -dk u 0 k 0 312 98 (42) S4-x), Cdk 3 K4 98 1 K 2 K 10. (42) 1 - 2 +) 3 k= olm 312 98 (42) ( 5 2K 한 것k K 3 VCS 5) 권 일 98 2 K 4k ㅋ + K (134 [ L - + 7 3 98 3. k 이나 1) (42) 9 312 2 (42) 98 16, 3 315 98 x 42 JAE x 16 3 315 11. 3135 42 45 NA2 Mnerefore, " 13136 AS 장이A 그 2