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Levart [38]
3 years ago
8

Find the surface area of the part of the sphere x2+y2+z2=81 that lies above the cone z=x2+y2−−−−−−√

Mathematics
1 answer:
TiliK225 [7]3 years ago
4 0
Parameterize the part of the sphere by

\mathbf s(u,v)=(9\cos u\sin v,9\sin u\sin v,9\cos v)

with 0\le u\le2\pi and 0\le v\le\dfrac\pi4. Then

\mathrm dS=\|\mathbf s_u\times\mathbf s_v\|=81\sin v\,\mathrm du\,\mathrm dv

So the area of S (the part of the sphere above the cone) is given by

\displaystyle\iint_S\mathrm dS=81\int_{v=0}^{v=\pi/4}\int_{u=0}^{u=2\pi}\sin v\,\mathrm du\,\mathrm dv=81(2\pi)\left(1-\dfrac1{\sqrt2}\right)=(162-81\sqrt2)\pi
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