Help with revision - area integrals

[Kawakaze]

I came across this question on a past paper and would appreciate some help. It is too hard for me at the minute.

The problem -

The volume of a body whose surface is formed on the underside
by the paraboloid z = x² + y² and bounded on top by the cone
z = 3-2(x² +y²)

(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.

 

[chiro]

Hey Kawakaze.

To get started, you should help us by telling us what you have thought about and what you have tried: even if it's only something partial and a few ideas rather than a fully fledged attempt.

 

[Kawakaze]

Well, I am guessing cylindrical coordinates would make the easiest calculations. But I really suck at the conversion.

 

[chiro]

In terms of integration and calculating integrals, we have a well developed way of finding integrals that are expressed in Cartesian space (i.e. R^n).

For the Riemann integral in R^n, we have well established theorems if we want to calculate integrals and if the integrals are in a simple enough form, then it is easier to use the R^n system since the mathematics is a lot simpler.

You can do calculations and stuff in curved geometries (and this is exactly what has to be done in things like General Relativity in physics as well as things in engineering and applied mathematics), but if it's in R^n and it's simple, then its a lot easier to deal with.

One question you might want to ask yourself is for these given objects (cone and paraboloid), is it straight-forward to do a volume calculation for these objects if we use calculus for normal three dimensions using the normal geometry?