題目 Problem 8 (10%). Evaluate the integral ∬SxydS\iint_S xydS∬SxydS, where the surface S={(x,y,z)∣x2+y2=1,x≥0,y≥0,0≤z≤1}S=\{(x,y,z)\mid x^2+y^2=1,x\ge 0,y\ge 0,0\le z\le 1\}S={(x,y,z)∣x2+y2=1,x≥0,y≥0,0≤z≤1}. 解答 解法一 以 θ,z\theta,zθ,z 參數化圓柱面：x=cosθ,y=sinθx=\cos\theta,y=\sin\thetax=cosθ,y=sinθ，其中 0≤θ≤π/20\le\theta\le\pi/20≤θ≤π/2、0≤z≤10\le z\le10≤z≤1，且 dS=dθ dz\mathrm{d}S=\mathrm{d}\theta\,\mathrm{d}zdS=dθdz。故 ∬Sxy dS=∫01∫0π/2cosθsinθ dθ dz=12.\iint_Sxy\,\mathrm{d}S =\int_0^1\int_0^{\pi/2}\cos\theta\sin\theta\,\mathrm{d}\theta\,\mathrm{d}z =\boxed{\frac12}.∬SxydS=∫01∫0π/2cosθsinθdθdz=21.
