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Problem: Find the mass M of a thin shell with density δ(x, y) = 1 + x2 + y2 occupying that part of the surface z = (x2 + y2)/2 lying below the plane z = 2.
Solution: We have
M = ∫S δ(x, y) dA = ∫S(1 + x2 + y2) ...
We compute the mass of a thin paraboloidal shell of given density by means of a double integral.