第1个回答 2013-03-07
x=sint,y=cost=(1-x^2)^(1/2),z=sin2t=2sintcost=2xy,dx=costdt,dy=-sintdt,dz=2cos2tdt,
∫c (y+sinx)dx =∫c ((1-x^2)^(1/2)+sinx)dx =1/2*arcsinx+1/2*x*(1-x^2)^(1/2)-cosx
=1/2*t+1/2*sint*cost-cos(sint)( t 从0到2π)= π
∫c (z^2+cosy)dy=∫c (4*y^2*(1-y^2)+cosy)dy=∫c (4*y^2-4y^4+cosy)dy
=4/3* y^3-4/5* y^5+siny=4/3* cost ^3-4/5* cost ^5+sin(cost) ( t 从0到2π)=0
∫c x^3dz=∫c sint ^3*2cos2tdt=∫c sint (1-cos2t)/2*2cos2tdt=∫c sint cos2tdt-∫c sint(cos2t)^2dt
=∫c sint cos2tdt-∫c sint(1-cos4t)/2*dt
=∫c sint cos2tdt-1/2*∫c sintdt+1/2*∫c sintcos4t)dt( t 从0到2π)=0
所以,原积分=π+0+0=π