Evaluate $\int_{\gamma}xy\ dx $ where $\gamma$ is the boundary of the square with vertices $(0,0),(1,0),(1,1),(0,1)$.
Now Green's Theorem says that $D$ be a bounded domain with piecewise smooth boundary $\delta D$ and $P(x,y)$ and $Q(x,y)$ be continuously differentiable functions on $D\cup {\delta D}$ then $$\int_{\delta D}P\ dx\ +Q\ dx\ \ =\ \ \int \int_{D}\left({{\delta Q}\over {\delta x}}-{{\delta P}\over {\delta y}}\right)dx\ dy$$
Now $Q$ is already $0$ here , we have $P(x,y)=xy$ . $${{\delta P}\over {\delta y}}=x$$ giving us ,$$-\int \int_D x\ dx\ dy\\=-\int_0^1 \int_0^1 x\ dx\ dy\\={{1}\over {2}} $$
Is it correct $?$
Actually I am not sure about this question because while evaluating the double integral I used limits as $(x\ varying\ 0\rightarrow 1)$ and $(y\ varying\ 0\rightarrow 1)$ and got same value on both sides but in a worked out example in the book about a triangle with vertices $(0,0),(1,0),(0,1)$
they did the calculation as $$\int_0^1\left[\int_0^{1-x}\ x\ dy\right]\ dx$$ i.e. as limits they used numbers for $x$ and functions for $y$ . So I do not understand why or when I should use numbers and when functions as limits for integration . Not functions actually it is said to be parametrization . I could not think of any parametrization for this square .
So what I indeed need is to know how to choose the limits for double integrals ,. Explanations are most welcome . Thanks.