$$
\begin{align}
\int_\Sigma\nabla\times F\cdot\mathrm{d}S
&=\int_\Sigma\left[
\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)\mathrm{d}y\,\mathrm{d}z
+\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)\mathrm{d}z\,\mathrm{d}x
+\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\mathrm{d}x\,\mathrm{d}y
\right]\\
&=\int_\Sigma\left[
\left(\frac{\partial P}{\partial y}\,\mathrm{d}y+\frac{\partial P}{\partial z}\,\mathrm{d}z\right)\mathrm{d}x