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- Gauss’s law says that the electric flux through any closed surface is equal to the total chargecontained in the closed surface divided by the permittivity of free space, E0Find the charge contained inside a cube with vertices at (±1, ±1, ±1) when E =< x, y,z >A solid E lies within the cylinder x2 + y2 = 9, below the plane z = 21, and above the paraboloid z = 9 − x2 − y2. (See the figure above.) The density at any point is proportional to its distance from the axis of the cylinder. Find the mass of E. Solution In cylindrical coordinates, the cylinder is r = and the paraboloid is z = , so we can write E = (r, ?, z) 0 ≤ ? ≤ 2?, 0 ≤ r ≤ 3, 9 − r2 ≤ z ≤ 21 Since the density at (x, y, z) is proportional to the distance from the z-axis, the density function is f(x, y, z) = K x2 + y2 = Kr where K is the proportionality constant. Therefore, from this formula, the mass of E is m = E K x2 + y2 dV = 2? 0 3 0 21 9 − r2 r dz dr d? = 2? 0 3 0 Kr2 dr d? = K 2? 0 d? 3 0 (12r2 + r4) dr = 2?K 3 0 =…