Question

1 R 12. Use the transformation T: u = -x and very to evaluate the integral [jx?dA where R is the region bounded on the xy-plane by the ellipse 9x + 4y = 36. . Let S be the image of Runder T on the uv-plane. Sketch regions and S. Set up the integral 7as an iterated integral of a function f(u, v) over region S. Use technology to evaluate the integral. Give the exact answer. R S Y

Answer 1

we can evaluate I xd transfor maton UX, using 12 / where R is Эivч ч by the equation 9x'+4416 2) gae² + 4 y2 -1 36 36 ) x² + 519 = 2) (+3) above of ey in figure is My (013) 10,32

the for limit part Now Draw areis ape Strip parallel qx² = 36-49² a = 4(9-y?) १ x2 2 (9-4) IV ? (9-44) X? I 6-y² 3 X-A1 Visay? '. limit of x is - Vigry to vo I regryz v za to y = 3 limit of y is y = - 3 and hare u = 1 a & Ś cy 2 J əlx,y) da 74 ον acuiv) oli ole Oy dy วบ ju

Now imge of 9x + 4y 16 mation " and ty trans for 9x +44 34 962474 4 (30 300+ 360 36 het me which is Also J: (6,9 ) O (414) 2 *Wall (Oia) llo) u

2 NOW I dudu R by the (o = 8s (1-62% Ss x da JS (20)? 3 Esl 4u? (6) du du figure = 11 24 4² du du u² du ar - - V -22 dre + (1.28233/2 dre 202 as - s 24 Vire 1 = 241 ( ) -1 -Vine -1 16 (1–20293/2 dre 3/2 du 32 j (1-0², 10²=t Let au are a dt

Aze dt 2 the dua {"l2 dt 2 It U20 21 tao a) t = 1 ci 고 32 g (1-tg/ dt Eth Įci stike.co-tole.de 36 5-1 16 -Go dE * (1-1) BC 16 S.) (B Bremen) = a meno 16 B/2 512 8/2 16 2/2 32 'krve 3! 16 x 3 8X8 2T