曲线积分∫(2xy^3-y^2cosx)dx+(1-2ysinx+3x^2y^2)dy,其中L为在抛物线2x=πy^2上

P=2xy³-y²cosx,Q=1-2ysinx+3x²y²
易验证：∂Q/∂x=∂P/∂y=6xy²-2ycosx
因此本题积分与路径无关,可自选积分路线
选从(0,0)到(π/2,1)的折线,
L1：y=0,x：0--->π/2
L2：x=π/2,y：0--->1
则原积分=∫L1 (2xy^3-y^2cosx)dx+(1-2ysinx+3x^2y^2)dy
+∫L2 (2xy^3-y^2cosx)dx+(1-2ysinx+3x^2y^2)dy
=0+∫[0--->1] [1-2y+(3π²/4)y²]dy
=y-y²+(π²/4)y³ |[0--->1]
=π²/4