原式=∫∫(xy+yz+xz)√[1+(αz/αx)²+(αz/αy)²]dxdy (S：x²+y²=2ax)
=∫∫(xy+yz+xz)√[1+(x/√(x²+y²))²+(y/√(x²+y²))²]dxdy
=√2∫∫(xy+yz+xz)dxdy
=√2∫dθ∫(r²sinθcosθ+r²sinθ+r²cosθ)rdr (做极坐标变换)
=√2∫dθ∫(sinθcosθ+sinθ+cosθ)r³dr
=(√2/4)∫(sinθcosθ+sinθ+cosθ)(cosθ)^4dθ
=(√2/4)∫[(1-2sin²θ+(sinθ)^4)cosθ+((cosθ)^5+(cosθ)^4)sinθ]dθ
=(√2/4)[sinθ-(2/3)sinθ+(1/5)(sinθ)^5-(1/6)(cosθ)^6-(1/5)(cosθ)^5]│
=(√2/4)(1-2/3+1/5+1-2/3+1/5)
=4√2/15.