计算对曲线积分∫z ds,其中C为螺旋线x=tcost,y=tsint,z=t(0≤t≤t0).
C：x=tcost,y=tsint,z=t；dx/dt=cost-tsint；dy/dt=sint+tcost；dz/dt=1；
[C]∫z ds=[C]∫t√[(cost-tsint)²+(sint+tcost)²+1]dt
=[C]∫t√[(cos²t-2tsintcost+t²sin²t)+(sin²t+2tsintcost+t²cos²t)+1]dt
=[C]∫t√(t²+2)dt=(1/2)∫√(t²+2)d(t²+2)=(1/2)(2/3)(t²+2)^(3/2)︱[0,to]=(1/3)[(t²o+2)^(3/2)- 2√2]