Calculate a line integral

Question

I need to calculate a line integral: $\int_\gamma (2x+1)dx+(3xy+1)dy$, where $\gamma$ is the surface of the traingle $OAB$ where $O(0,0)\ \ \ A(3,-3)\ \ \ B(-6,-6) $.
I tried to solve this by parametrizing the lines $AB$, $AO$ and $OB$, but it turns out to be a hard to compute integral, so I was wondering if you can help me here. Thank you

Answer 1

Let the interior of the triangle be $A$.

By Green's theorem, \begin{align*} \oint_{\gamma} (2x+1) \,dx + (3xy+1) \, dy &= \iint_A \left(\frac{\partial}{\partial x} (3xy+1) - \frac{\partial}{\partial y}(2x+1)\right)\,dx\,dy\\ &= \iint_A 3y \, dx \, dy. \end{align*} You can now parametrise the interior of the triangle, and this integral is much easier to compute than working directly.