Double integral on a compact subset Calculate:
$$\int \int _D \left(6x+2y^2 \right) dxdy$$
where D is a compact subset of $\mathbb{R}^2$ enclosed by a parabola $y=x^2$ and a line $x+y=2$.
How to find that, how to find the limits of integration in this case? I think then i will manage to calculate the whole integral.
I think the area is the following (enclosed by points ABC):


First $x^2 + x - 2 = 0 \iff x = 1$ or $x = -2$. Then $$\iint_D (6x^2+ 2y^2) dA =\int_{-2}^1\int_{x^2}^{2-x} (6x  +2y^2)dydx$$
