Surface s is a part of the paraboloid z=4−x2−y2 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 r→u,r→v, and r→u⋅r→v. c) Find the area of the surface

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Surface s is a part of the paraboloid z=4−x2−y2 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 r→u,r→v, and r→u⋅r→v.  c) Find the area of the surface

Viktor Wiley · Accepted Answer

Given :  z=4−x2−y2, z=0  a) x=ucos⁡v,y=usin⁡v,z=4−u2cos2v−u2sin⁡v                         z=4−u2  r(u,v)=&amp;lt;ucos⁡v,usin⁡v,4−u2&amp;gt;  The domain of 0≤u≤2                0≤v≤2π  b) ru=&amp;lt;cos⁡v,sin⁡v,−2u&amp;gt;    rv=&amp;lt;−usin⁡v,ucos⁡v,0&amp;gt;  ruxrv=[ijkcos⁡vsin⁡v−2u−usin⁡vucos⁡v0]  =i(0+2u2cos⁡v)−j(−2u2sin⁡v)+k(ucos2v+usin2v)=&amp;lt;2u2cos⁡v,2u2sin⁡v,u&amp;gt;  c) Surface Area =∫02π∫02|ruxrv|dudv  |ruxrv|=4u2cos2v+4u4sin2v+u2=8u4+u2  =u1+8u2  A=∫02π∫02u1+8u2dudv  1+842=f  =∫02πdv∫fdf16  16udu=df

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.

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