Surface s is a part of the paraboloid z=4-x^2-y^2 that lies above the plane z=0.(6+7+7=20pt) a)Find the parametric equation vecr(u,v) of the surface with polar coordinates x=ucos(v), y=usin(v) and find the domain D for u and v.

OlmekinjP 2021-09-12 Answered

Surface s is a part of the paraboloid z=4x2y2 that lies above the plane z=0.(6+7+7=20pt)

a) Find the parametric equation r(u,v) of the surface with polar coordinates x=ucos(v),y=usin(v) and find the domain D for u and v.

b) Find ru,rv, and rurv.

c) Find the area of the surface

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Expert Answer

Viktor Wiley
Answered 2021-09-13 Author has 84 answers
Given : z=4x2y2, z=0
a) x=ucosv,y=usinv,z=4u2cos2vu2sinv
z=4u2
r(u,v)=<ucosv,usinv,4u2>
The domain of 0u2
0v2π
b) ru=<cosv,sinv,2u>
rv=<usinv,ucosv,0>
ruxrv=[ijkcosvsinv2uusinvucosv0]
=i(0+2u2cosv)j(2u2sinv)+k(ucos2v+usin2v)=<2u2cosv,2u2sinv,u>
c) Surface Area =02π02|ruxrv|dudv
|ruxrv|=4u2cos2v+4u4sin2v+u2=8u4+u2
=u1+8u2
A=02π02u1+8u2dudv
1+842=f
=02πdvfdf16
16udu=df

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