I've just done a problem on parameterized volumes:

"Find a parameterization for the volume which lies below the cone z=r, above the x-y plane, and inside the cylinder x<sup>2</sup>+y<sup>2</sup>=1"

What I did was:

Ψ(r,θ)=(rcosθ,rsinθ,r)

0<θ<2π, 0<r<1 (<is less than or equal to)

but the answer was:

Ψ(r,θ,t)=(rcosθ,rsinθ,tr)

0<θ<2π, 0<r<1, 0<t<1

I'm wondering why is it necessary to parameterize with t, since r is already between 0 and 1?

For a parameterized volume in R<sup>3</sup>, will you always have three parameters, as opposed to having two to represent a surface?...you might even be able to parameterize the volume with one parameter thoguh might you?

Doesparameterizinga volume give any easier way to calculating the actual value of the volume (say r is in metres)? For this question it would be pretty easy to calculate the volume normally, but for more complicated parameterized volumes perhaps there is a method that can be used to speed things up/make things easier?...just something I was wondering about.

Thansk alot