- #1

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The problem -

The volume of a body whose surface is formed on the underside

by the paraboloid z = x

^{2}+ y

^{2}and bounded on top by the cone

z = 3-2(x

^{2}+y

^{2})

(a) Explain which coordinate system should be used in order to describe

the shape easily, and transform the surfaces bounding the shape into

this coordinate system (if necessary).

(b) Find the equation of the curve of intersection of the two surfaces, and

describe it.

(c) Explain the order of integration of the variables to facilitate the

formation of the integral. State without using symbols or too many

numbers what the limits will be for each integration, and describe the

shape formed after each integration, starting with the basic cuboid

element.

The density of the material is σ(1 + ρ) where σ is a constant, and ρ is the

distance from the z-axis.

(d) Set up the volume integral to find the mass of the volume, M, and

determine the value of the mass in terms of σ.

(e) Set up the volume integral to find the moment of inertia of the volume

about the z-axis, and determine this moment of inertia in terms of M.

(f) Set up the integral to find the moment of inertia about the x-axis, but

do not evaluate it.

(g) Will the moment of inertia about the y-axis be the same as, or

different to, that about the x-axis? Briefly justify your conclusions.