3. Let f(x, y, z) = x(1 x y 2), where s 0 is a parameter. a) [3 marks] Sketch the region of integration for the following triple integral, [ [ [ f (x, y, z) dV = SS S = S. So 1 1-x 1-x-y f(x, y, z) dz dy dx. b) [3 marks] Evaluate the repeated integral from a) for the special case when S = 0. c) [3 marks] Evaluate the repeated integral from a) for the special case when S = = 1. d) [2 marks] The change of variables defined by the relations x+y+z = U, y + z = uv, z = uvw. maps the region defined by 0 u 1, 0 v 1, 0 w 1 to the region (you DO NOT need to prove this). Express x, y, z as functions of u, v, w. e) [4 marks] Show that the Jacobian determinant of the mapping from d) equals u2v (and so conclude that the mapping F is one-to-one on the interior of F). f) [5 marks] By setting up an appropriate triple integral using the variables u, v, w, show that ] [ f ( x, y, z ) d V = 1 (s+4)(s+3)(s + 2)(s + 1) *