My question has to do with picking the correct limits for integration. I thought I had it figured out well, but I had an interesting issue with a homework problem. The problems were about Green's Theorem. As you know the following is the theorem for Flux integrals $\int \int_R \frac{\partial M}{\partial x} + \frac{\partial N}{\partial y} dx dy$. This is a homework problem which I worked but came to the wrong answer. The path $C$ along the curve $y=x^2, (0,0) \rightarrow (1,1)$ and then $x=y^2, (1,1) \rightarrow (0,0)$. Because the picture was drawn this way, I set up my integral like this
$$ \begin{array}{rcl} \mathbf{F} & = & \langle xy+y^2, x-y \rangle \\ \end{array} \\ \begin{array}{cc} M = xy+y^2 & \frac{\partial M}{\partial x} = y \\ N = x - y & \frac{\partial N}{\partial y} = -1 \\ \end{array} \\ \int_{1}^{0}\int_{y^2}^{\sqrt{y}} \frac{\partial M}{\partial x} + \frac{\partial N}{\partial y} dxdy = \int_{1}^{0}\int_{y^2}^{\sqrt{y}} y-1 dxdy $$
This integral resulted in the correct answer by magnitude but the wrong sign. I found that by using this integral
$$ \int_{0}^{1}\int_{y^2}^{\sqrt{y}} y-1 dxdy $$
Yeilded the correct answer. I used the limits I did originally because I was picturing "a particle" or something similar moving along the path in the path indicated. However, I got the wrong answer because of my choice of limits. Why did I get the limits incorrect, or did I? As I write this question, and think through what I've learned from past courses, should I have negated the integral? Like this?
$$ - \int_{1}^{0}\int_{y^2}^{\sqrt{y}} y-1 dxdy $$
Thanks,
Andy