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

GrareeCowui 2022-01-23 Answered
Let S denote the solid enclosed by x2+y2+z2=2z and z2=x2+y2. What is the length of of the curve determined by (x,y,z): x2+y2+z2=2z and z2=x2+y2? What is the surface area of S?
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Expert Answer

primenamaqm
Answered 2022-01-24 Author has 12 answers
We have two surfaces:
x2+y2+z2=2z and z2=x2+y2
The loci of the intersection of the surfaces is thus that of the simultaneous solution, thus:
(z2)+z2=2zz2z=0z=0,1
Leading to two loci:
{z=0z=1{x2+y2=0  a circle of radius  0x2+y2=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 Pcircle=2πr) is:
L=2π
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