Question

高等数学第二曲线积分问题

高等数学第二曲线积分问题求写下解题过程，最后答案已经给出了

Answer 1

3. 用格林公式， I = ∫∫<D>(∂q/∂x-∂p/∂y)dxdy = ∫∫<D>dxdy = 1.
5. P = -y/√(x^2+y^2) = -sint, Q = x/√(x^2+y^2)= cost, R = 0
I = ∫<0, 2π>[(-sint)(-asint) + costacost]dt = a∫<0, 2π>dt = 2πa
6. 设 x = cost, y = sint, z = 2-cost+sint,
I = ∫<0, 2π>[(2-cost)(-sint) + (2cost-sint-2)cost + (cost-sint)(sint+cost)]dt
= ∫<0, 2π>[3(cost)^2-(sint)^2-2sint-2cost]dt
= ∫<0, 2π>[1+2cos2t-2sint-2cost]dt
= [t+sin2t+2cost-2sint]∫<0, 2π> = 2π