Question

Multivariable calculus question?

Let S denote the solid enclosed by `x^2+y^2+z^2=2z` and `z^2=x^2+y^2`.

What is the length of of the curve determined by (x,y,z): `x^2+y^2+z^2=2z` and `z^2=x^2+y^2` ?

What is the surface area of S?

Answer 1

the length of of the curve is `2pi`

Explanation:

For the First Part:

We have two surfaces:

`x^2 + y^2 + z^2 =2z` and `z^2 = x^2 + y^2`

The loci of the intersection of the surfaces is thus that of the simultaneous solution, thus:

`(z^2) + z^2 =2z => z^2-z = 0 => z=0,1`

Leading to two loci:

`{ (z=0), (z=1) :} => { (x^2+y^2=0, " a circle of radius "0), (x^2+y^2=1," a circle of radius "1) :}`

If we examine the surfaces, then the first solution is that of a single (tangency) point, and thus the second solution is the sought solution, as such the length of the curve (using `P_("circle")=2pir`) is:

`L = 2pi`