计算∫L（x^2+2xy）dy,其中L是椭圆由点A（a,0）经点B（0,b）到点C（-a,0）的弧段L在积分号下面

Question

计算∫L（x^2+2xy）dy ,其中L是椭圆 由点A（a,0）经点B（0,b）到点C（-a,0）的弧段
L在积分号下面

Answer 1

{ x = acost { y = bsint
{dx = - asint dt { dy = bcost dt
∫_[L] = ∫_[A→B] + ∫_[B→C]
∫_[L] (x² + 2xy) dy
= ∫[0→π/2] [a²cos²t + 2(acost)(bsint)](bcost) dt + ∫[π/2→π] [a²cos²t + 2(acost)(bsint)](bcost) dt
= ∫[0→π/2] [a²bcos³t + 2ab²sintcos²t] dt + ∫[π/2→π] [a²bcos³t + 2ab²sintcos²t] dt
= (2/3)ab(a + b) + (2/3)ab(b - a)
= 4ab²/3