GrareeCowui
2022-01-23
Answered

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?

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primenamaqm

Answered 2022-01-24
Author has **12** answers

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:

$\left({z}^{2}\right)+{z}^{2}=2z\Rightarrow {z}^{2}-z=0\Rightarrow z=0,1$

Leading to two loci:

$\{\begin{array}{c}z=0\\ z=1\end{array}\Rightarrow \{\begin{array}{cc}{x}^{2}+{y}^{2}=0& \text{}\text{a circle of radius}\text{}0\\ {x}^{2}+{y}^{2}=1& \text{}\text{a circle of radius}\text{}1\end{array}$

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${P}_{\text{circle}}=2\pi r$ ) is:

$L=2\pi$

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

Leading to two loci:

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

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