Question

Points Use the given transformation to evaluate the integral SLO (20x + 18), where is the paralelogram with vertices (-), 12), (3, -12), (6, -9), and (0, 13); } + 7). 4050

Answer 1

R 2 6 Gyver Integral, SS (20x+1SH) LA Where R is the parallelogram with vertices (-3, 12), (3,-12), (6,~9) and (0,15) Gulver transformation 2-ţ(4+0) y = $(V-40) The Region of integration in xy more és a parablelogram AB CD, where A (-3,12), B (3,-12), C (69) and Dl015) in xyplore Pogint in we have plate 2 = 5 (470) and y = 1 (V-44) About utr=580 V-44= 54- How solving couro 8 2 u+x=50 x-yu = sy >5u=5X-sy » Bu=f(x-4) u=x-Y How putting value of u=x-Y in egun ♡ x-y+v=5X - V=5X-aty V=4&ty

Point en u=x-Y V=4&ty corresponding Point ér (u, v) hare/16 X-y Hare A (-3,12) u=-3-12 u=-15 V=4(-3)+12 V=~12+12=0 A'C-15,0) X=-3, y=12 u=3-(42) - U=15 V=4 (3)-12 v=12-12=0 B' (15,0) 13 (3,-12) 9=3, 4342 c 16,-9) a=6, y=-g u= 6-ng) ♡ u = 15 V=416-9 V=24-9=15 e (15,15) D10, 15) aco, y=15 D'-15,15) 4 = 6-15 u=~15 v=ylotis &v=o415515 He new region of integration în ur-plare rectangle A' B'c' d' in uv-plane و لد

Islo, is) A 14 (-32) g7 16 3R + + + -12 og -6 - 3 3. 6 9 + -6 Rough Sketch 469) 1 - 9 7-12 13 (312) Va is V S 15,15 T27 (415,159) u=1s R gt tool=15 6- + + a wt 7u 6 B' 9 12 15 (15,5) 37. A + ..-15 12- -TS 12-9 -6 -3 11-15, 0) +-6 +-g - 12 +-15 t-3 Rough Sketch

2 su ди ar Gl- ax du dy du dr sy ar 1 +4 51 5 We have, X= 14 lut) 4- (-40) 13 ay=-Y O2 = 24 = yes J = 2(x, y) da 15 V əlur) -Yg Vs 25 ( - ) 25 25 285 y J = 2(x, y) əlur) de dy 15l du dr = ( / J du dr SS. (208-+ 154) da = 8301200+154) de dy (20x(446)41544_V_u4 15 Yo 2013 (4+VD+L&X(1-44) 5) dudy u=15 Y=0 Eg s 34444v)+317-4013 () du dv 4 =-15 roo (44+4V+38-1249 de dr u=-1s ro - $ Surmis (7V-8 u) dr du 15 015 IS 15 Is f's 11 V=0

11 1015 "782-8 uv 15 du 5 1 / u=45 ivo GN - , $ 7x1532 - Bux1sf du 7x15 - 8u) du 15 - 375 urls 15 = 3 els 105 – 8 u) du u=-15 105 u educere uaris = 3 Marls = 3 Чu2 2 [1059.442's 83 (105X15 – u X(153*) – (105*2195)_466-153) = 3 S 105X15 - 4x19x+105x15 +4x5} = 3 xXx 105x15 SXS = 105 x 45=4725 SS (201+ ISY) da 2 4725! Answer