计算对面积的曲面积分i=∫∫∑(xy+yz+xz)ds.其中∑是圆锥面z=√(x^2+y∧2)被

Question

如题所述

Answer 1

简单计算一下即可，答案如图所示

Answer 2

∵dS=√[1+(αz/αx)²+(αz/αy)²]dxdy
=√[1+(x/z)²+(y/z)²]dxdy
=√2dxdy
∴原式=∫dθ∫(r²sinθcosθ+r²sinθ+r²cosθ)rdr (做极坐标变换)
=4a^4∫(sinθcosθ+sinθ+cosθ)(cosθ)^4dθ
=4a^4∫[sinθ(cosθ)^5+sinθ(cosθ)^4+cosθ(1-2sin²θ+(sinθ)^4)]dθ
=4a^4[(-1/6)(cosθ)^6+(-1/5)(cosθ)^5+sinθ-(2/3)sin³θ+(1/5)(sinθ)^5]│
=4a^4(1-2/3+1/5+1-2/3+1/5)
=(64/15)a^4本回答被网友采纳