Line integrals

[mcwang719]

Find the circulation \(\displaystyle (\oint {F \cdot dr} )\) around C, the boundary of the region \(\displaystyle 0 \le y \le x^2\) and \(\displaystyle 0 \le x \le 1\). The fluid velocity vector field is \(\displaystyle F(x,y) = \left\langle {xy,x^2 + \ln (y + 5)} \right\rangle\) Can anyone help or guide me through this problem. This whole chapter on line integrals is confusing me. thanks!!!!!!!!!!!

 

[Unco]

If you're familiar with Green's Theorem, I suggest you use it here. If you're not, parameterise the curve as usual. For instance, and assuming counterclockwise orientation, the parabolic part could be parameterised by r(t) = (t, t^2), where t goes from 0 to 1.