Problem 1 if d is the unit sphere centered at 0 0 0

Question

Problem 1: If D is the unit sphere centered at (0, 0, 0), evaluate

Problem 2: The thickness of a triangular metal plate varies as f (x, y) = (1 + xy) cm. Find the average thickness of the plate given in the diagram below.

Problem 3: Find the centre of mass of a solid hemisphere radius 1 centered at (0, 0, 0) for z ≥ 0, if the mass density per unit volume μ is constant.

Problem 4: If the density of material inside the solid cone

z = 2 √x² + y², z ≤ 4

varies as

μ (x, y, z) = 5 - z,

Find the moment of inertia about the z-axis.