i have to Integrate the following real vectorfield over the following path: Let F be: $$ F(x,y) = (x^2(y-1)^2, x(y-1)) $$ First lets integrate the direct path between those points $$ \displaystyle\int_{(0,1)}^{(0,0)} F = a $$ and then between these: $$ \displaystyle\int_{(0,0)}^{(1,0)} F = b. $$ Now take the Integral: $$ \displaystyle\int_{j} F = c. $$ so that j is the parabolic way such that $$y = 1-x^2$$ from $(0,1)$ to $(1,0)$. I want to calculate the difference of $a$ and $b$ combined with $c$: $(a+b)-c$.
I calculated that $c$ should be $c=16/35$ and for $a+b$ i got $1/4$. Is that correct and how can i integrate those using greens theorem?
Hoping for help! (: 
So, you have established that $$ \int_{\gamma} F dr = \int_{\gamma_1} F dr+\int_{\gamma_2} F dr+\int_{\gamma_3} F dr = \cdots = \frac{22}{105} $$
because
$$ \int_{\gamma_1} F dr = \int_1^0 (t^2,-t)\cdot (1,0) dt = [t^3/3]_1^0 = -\frac 13 $$
$$ \int_{\gamma_2} F dr = \int_0^1 (0,0)\cdot (1,0) dt = 0 $$
$$ \int_{\gamma_3} F dr = \int_0^1 (t^2(1-t^2-1)^2,t(1-t-1))\cdot (1,-2t) dt = \frac{19}{35} $$
Using Green's theorem, you get the same result...
$$ - \int_{\gamma} (P dx + Q dy) = \iint_{D} \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} dx dy = \int_0^1 \int_0^{1-x^2} (y-1 - 2(y-1)x^2) dy dx $$