4. For each of the following, evaluate Is w using Stokes's Theorem or Gauss's Theorem.

(a) S is topological boundary of cylindrical solid y2 + z2 ~ 9, ° ~ x ~ 2, with outward-pointing normal, and w = xy dy dz + (x2 - z2) dz dx + xz dx dy.

(b) S is the truncated cylinder x2 + z2 = 8, ° ~ y ~ 1, with outward-pointing normal, and w = (x - 2z) dy dz - y dz dx.

(c) S is the topological boundary of R = [0, 7r /2] x [0, 1] x [0,3], with outwardpointing normal, and w = eY cos x dy dz + x2 Z dz dx + (x + y + z) dx dy.

(d) S is the intersection of the elliptic cylindrical solid 2x2 + Z2 ~ 1 and the plane x = y, with normal that points toward the positive x axis, and w =

x dy dz - y dz dx + sin y dx dy.