I = ∮x^2ydx+y^2xdy =∫∫(y^2-x^2)dxdy =∫dt∫r^2[(sint)^2-(cost)^2]rdr=∫(-cos2t)dt[r^4/4] =∫4(cost)^4(-cos2t)dt= -2∫(1+cos2t)^2(cos2t)dt= -2∫[cos2t+2(cos2t)^2+(cos2t)^3]dt= -2∫(cos2t+1+cos4t)dt ...
利用格林公式计算
I = ∮x^2ydx+y^2xdy =∫∫(y^2-x^2)dxdy =∫dt∫r^2[(sint)^2-(cost)^2]rdr=∫(-cos2t)dt[r^4/4] =∫4(cost)^4(-cos2t)dt= -2∫(1+cos2t)^2(cos2t)dt= -2∫[cos2t+2(cos2t)^2+(cos2t)^3]dt= -2∫(cos2t+1+cos4t)dt ...