对坐标的曲线积分：要求详细步骤

Answer 1

有两个方法：
第一：格林公式
x² + y² = 2y => x² + y² - 2y + 1 = 1 => x² + (y - 1)² = 1
圆圈半径为1，面积 = π(1)² = π
令P = xy² + 2y，Q = x²y
∂Q/∂x = 2xy，∂P/∂y = 2xy + 2
∮_L (xy² + 2y)dx + x²ydy
= ∫∫_D (∂Q/∂x - ∂P/∂y) dxdy
= ∫∫_D [2xy - (2xy + 2)] dxdy
= - 2∫∫_D dxdy
= - 2 · D，D表示的就是圆圈的面积
= - 2π

第二：参数方程
令x = cost，y = sint
dx = - sint dt，dy = cost dt，t∈[0，2π]
∫_L (xy² + 2y)dx + x²ydy
= ∫(0-->2π) [(costsin²t + 2sint)(- sint) + (cos²tsint)(cost)] dt
= ∫(0-->2π) (sin²tcost - 2sin²t + sintcos³t) dt
= ∫(0-->2π) sin²t d(sint) - ∫(0-->2π) (1 + cos2t) dt - ∫(0-->2π) cos³t d(cost)
= [(1/3)sin³t - (t + 1/2 · sin2t) - (1/4)cos⁴t] |(0-->2π)
= (- 1/4 - 2π) - (- 1/4)
= - 2π