Question

求19题过程🙏🏻🙏🏻㈻

求19题过程🙏🏻🙏🏻🙏🏻

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Answer 1

19.作变换x=Rsinucosv,y=Rsinusinv,z=Rcosu,则
x'u=Rcosucosv,x'v=-Rsinusinv,
y'u=Rcosusinv,y'v=Rsinucosv,
z'u=-Rsinu,z'v=0,
E=x'u^2+y'u^2+z'u^2=R^2,
G=x'v^2+y'v^2+z'v^2=R^2(sinu)^2,
F=x'ux'v+y'uy'v+z'uz'v=0,
∴√(EG-F^2)=R^2sinu,
由z>=√(x^2+y^2),得cosu>sinu,0<=u<=π，
∴0<=u<=π/4.
I=∫<0,2π>dv∫<0,π/4>R^2(sinu)^2*R^2sinudu
=πR^4∫<1/√2,1>(1-t^2)dt(t=cosu)
=πR^4*[t-t^3/3]|<1/√2,1>
=πR^4*(2/3-√2/4).