Integral of $f(x)$ of region between $y=0$ and $y=4-x^2$ The limits of the integral
$$\iint 2xy\,dy\,dx$$
in the region between
$$\left(4-x^2\right)$$ and $$y=0$$ are 
$$-2\leq x\leq2$$
and 
$$0\leq y\leq4-x^2$$
Is this correct?

Yes, indeed.   Because $(-2,0)$ and $(2,0)$ are the two intercepts of the curves, then $\{(x,y): x\in[-2;2], y\in[0;4-x^2]\}$ is the region enclosed by those curves.   Thus the definite integral you seek is:
$$\int_{-2}^2\int_0^{4-x^2} 2xy \operatorname d y\operatorname d x$$
Continue from here.
