Use the given transformation to evaluate the integral. double integral 9xy dA R

Question

Use the given transformation to evaluate the integral. double integral 9xy dA R , where R is the region in the first quadrant bounded by the lines y = 2/ 3 x, and y = 3x, and the hyperbolas xy = 2 /3 and xy = 3; x = u/v, y = v

Explanation / Answer

Let x = u/v, y = v. So, y = x ==> v = u^(1/2), and y = 2x/3 ==> v = (2/v2) u^(1/2) and xy = 1 ==> u = 1, and xy = 2/3 ==> u = 2/3. Moreover, ?(x,y)/?(u,v) = |1/v -u/v^2| |0 1| = 1/v. So, ??R 9xy dx dy = ?(u = 1 to 2/3) ?(v = u^(1/2) to (2/v2) u^(1/2)) 9u * (1/v) dv du = ?(u = 1 to 2/3) 9u ln v {for v = u^(1/2) to (2/v2) u^(1/2)} du = ?(u = 1 to 2/3) 9u [ln((2/v2) u^(1/2)) - ln(u^(1/2))] du = ?(u = 1 to 2/3) 9u ln(2/v2) du = (9/2)u^2 ln(2/v2) {for u = 1 to 2/3} = (81/4) ln(2/v2).

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