Solved: Open and closed boxes Consider the region R

Chapter 13, Problem 62

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Open and closed boxes Consider the region R bounded by three pairs of parallel planes: ax + by = 0, ax + by = 1, cx + dz = 0, cx + dz = 1, ey + fz = 0, and ey + fz = 1, where a, b, c, d, e, and f are real numbers. For the purposes of evaluating triple integrals, when do these six planes bound a finite region? Carry out the following steps. a. Find three vectors n1, n2, and n3 each of which is normal to one of the three pairs of planes. b. Show that the three normal vectors lie in a plane if their triple scalar product n1 # 1n2 * n32 is zero. c. Show that the three normal vectors lie in a plane if ade + bcf = 0. d. Assuming n1, n2, and n3 lie in a plane P, find a vector N that is normal to P. Explain why a line in the direction of N does not intersect any of the six planes and therefore the six planes do not form a bounded region. e. Consider the change of variables u = ax + by, v = cx + dz, w = ey + fz. Show that J1x, y, z2 = 01u, v, w2 01x, y, z2 = -ade - bcf. What is the value of the Jacobian if R is unbounded?