Homework SourseLet R be the region bounded by the graphs xy1xy2 2x3y2 2x3y5
Let R be the region bounded by the graphs x+y=1,x+y=2, 2x-3y=2, 2x-3y=5. Use the change of variables x=1/5(3u+v) and y=1/5(2u-v) to evaluate the double integral(2x-3y)dA
Solution
x = (1/5)(3u + v), y = (1/5)(2u - v) <==> u = x + y and v = 2x - 3y. So, R transforms to S: u in [2, 4], v in 3, 5]. Since the Jacobian ?(x,y)/?(u,v) = |3/5 1/5| |2/5 -2/5| = -8/25, which has absolute value 8/25, we obtain ??R 10 dA = ??s 10 * (8/25) dA\' = 16/5 * (Area of S) = 16/5 * (4 - 3)(5 - 3) = 64/5.
