求∮L=(x^2;+y^2;)ds=,其中L是曲线:x^2+y^2=2y

∮L=(x^2;+y^2;)ds=,其中L是曲线:x^2+y^2=2y
2025-05-09 13:20:35
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回答1:

x = rcosu, y = rsinu
x^2+y^2 = r^2 = 2rsinu ==> r = 2sinu
曲线 x^2+(y-1)^2 = 1 是一个单位圆,圆心在(0,1).
∮L = ∫[0,pi] {∫[0, 2sinu] r^2 rdr} dθ
= ∫[0,pi] 4(sinu)^4 dθ
= ∫[0,pi] (1-cos2u)^2 dθ
= = ∫[0,pi] (1-2cos2u + (1/2)(1-cos4u)) dθ
= 3pi/2