Integral of $f(x)$ of region between $y=0$ and $y=4-x^2$

Question

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?

Answer 1

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:

$$\int_{-2}^2\int_0^{4-x^2} 2xy \operatorname d y\operatorname d x$$

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