Solved: Compound surface and boundary Begin with the

Chapter 14, Problem 37

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Compound surface and boundary Begin with the paraboloid z = x2 + y2, for 0 z 4, and slice it with the plane y = 0. Let S be the surface that remains for y 0 (including the planar surface in the xz-plane) (see figure). Let C be the semicircle and line segment that bound the cap of S in the plane z = 4 with counterclockwise orientation. Let F = 82z + y, 2x + z, 2y + x9. a. Describe the direction of the vectors normal to the surface that are consistent with the orientation of C. b. Evaluate 4S 1_ * F2 # n dS. c. Evaluate AC F # dr and check for agreement with part (b). z _ x2 _ y2 z C y x S 4