Green theorem: compare {xy-(x^2)} dx+)x^2)ydy} over the triangle bounded lines y=0,x=1,y=x and verify by Green's theorem

[iron123]

compare {xy-(x^2)} dx+)x^2)ydy} over the triangle bounded lines y=0,x=1,y=x and verify by Green's theorem
how to solve this

 

[mario99]

compare {xy-(x^2)} dx+)x^2)ydy} over the triangle bounded lines y=0,x=1,y=x and verify by Green's theorem
how to solve this

You have:

[imath]\displaystyle \oint_{C} \left(xy - x^2\right) \ dx + x^2y \ dy[/imath]

If you have a closed curve, Green's theorem helps you to integrate with respect to the area bounded by this curve, instead of integrating over the curve which needs parameterizing [imath]x[/imath] and [imath]y[/imath].

In other words, Green's theorem says:

[imath]\displaystyle \oint_{C} \left(xy - x^2\right) \ dx + x^2y \ dy = \iint\limits_{R} \left[\frac{\partial}{\partial x}\left(x^2 y\right) - \frac{\partial }{\partial y}\left(xy - x^2\right)\right] \ dx \ dy [/imath]

Solving this new integral is straightforward.