A vector field $\mathbf{G}(\mathbf{r}) = yz\mathbf{i}+\mathbf{j}+x^2\mathbf{k}$ fill all the space.
Evaluate the area integral $I=\int\mathbf{G}(\mathbf{r})\cdotp d\mathbf{A}$ over the rectangle in the $(x,y)$ plane with corners $ (0,0,0), (1,2,0), (1,0,0), (0,2,0). $
What I have tried so far:
$I=\int (yz\mathbf{i}+\mathbf{j}+x^2\mathbf{k})\cdotp (dx\mathbf{i}\cdotp dy\mathbf{j})$
I want to try and calculate the integral in terms of $x$ and have parametrised the diagonal from $(1,0,0)$ to $(0,2,0)$ as $y=2-2x$.
$dy=-2dx$
$I=\int ((2-2x)z\mathbf{i}+\mathbf{j}+x^2\mathbf{k})\cdotp (dx\mathbf{i}\cdotp -2dx\mathbf{j})$
I am not sure how to parametise $z$ in terms of $x$. Unless there is a different approach to this question? If possible diagrams would be helpful.
$$I=\int yz\mathrm{d}y\mathrm{d}z+\mathrm{d}z\mathrm{d}x+x^2\mathrm{d}x\mathrm{d}y=\int_0^1\mathrm{d}x\int_0^2 x^2\mathrm{d}y=2/3$$