Let I = f(x² - y²) dx dy, where D = {(x, y) : 3 ≤ xy ≤ 4,0 ≤ x - y ≤ 3, x ≥ 0, y ≥ 0} Show that the mapping u = xy, v = x - y maps D to the rectangle R = [3, 4] × [0,3]. (a) Compute (x, y)/(u, v) by first computing (u, v)/8(x, y). (b) Use the Change of Variables Formula to show that I is equal to the integral of f(u, v) = v over R and evaluate. 8(z,y) = (b)I =

b) Compute f. F. dr where c is the boundary of the square [0, 1] × [0, 1] where F = (–y², x²).

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Let F = (r-y, z+y+3, z2) and S = {(r, y, z) : y = 1– r - z, y 2 0} oriented in the posItIve y-direction. Compute F.dS.