Question

The region R is bounded by the following curves. x2 + y2 = 4 x2 + y2 = 9 x2 - y2 = 1 x2 - y2 = 4 (a) Find a change of variables such that the transformed region is a rectangle in the uv-plane. u= V= (b) Draw a picture of S, the transformation of R into the uv-plane. Y υ 3 10 8 2 6 R 4 1 N х и 4 6 8 10 N (c) Evaluate Strºy + xy*dAgy

Answer 1

banded by the following [ Given the Region R is lowes get ga =4 , g4 g²-9, plugtel, rugas y ghty 2 and Va phry2 let by then the transformed region are bounded the Curves MzY Mv plane which is clearly a Rectangle M = 9 Val 7

DATE (622) 9 do 4 124 S V의 My Id Axy R If (x3y + a Scalytag] dedy R. I ay Cotty dedy Let pity - 4 gange av R ten de du dady

- Now We know hat dreu) Now 2x 2y - 4xy - 4xy 8ry d العمل على Bay i da dyz I deu du say The transformed Integral is If ay Cx²ty") dedja log u deed v R S 8 $ tls u du du V24 429 ㅗ s udell du Val M2 Y 8

DATE و 8 [3] // د ( 2. d 2 - ] [-] 3 - 373 را - و - (( )] b1ک 16,