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Computing areas and volumes using multiple integrals.

(1) Find the volume of the solid bounded by the paraboloid x2 + y2 = 2z, the plane
z = 0 and the cylinder x2 + y2 = 9.
(2) Find the volume of the region in the first octant bounded by x + 2y + 3z = 6.
(3) Find the area of the solid that is bounded by the cylinders x2+z2 = r2 and y2+z2 =
r2.
(4) Find the volume enclosed by the surfaces z = 8 − x2 − y2 and z = x2 + 3y2.

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Computing areas and volumes using multiple integrals is investigated. The solution is detailed and well presented.

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