计算∫xyds,其中为从(0,0)到(2,0)的上半圆弧:x^2+y^2=2x

Question

计算∫xyds ,其中 为从(0,0)到(2,0)的上半圆弧:x^2+ y^2 =2x

Answer 1

用参数方程：x² + y² = 2x ==> (x - 1)² + y² = 1
{ x = 1 + cost { dx = - sint dt
{ y = sint { dy = cost dt
∫L xy ds
= ∫(0→π) (1 + cost)(sint) dt
= ∫(0→π) (sint + sintcost) dt
= - [ cost ]：(0→π)
= - [ - 1 - 1]
= 2