I am having problems parameterizing these integrals:
$$\int_A{\frac{x}{1+x^2+y^2}}\mathrm{d}x\mathrm{d}y$$ for $A = \mathbb R^2 \bigcap \,\{y \ge 0\}$
and the volume of $M = \{(x, y, z) \in \mathbb R^3 : 0 \le y \le 1, \; 1 \le x^2 + z^2 \le 4,\; 0 \le z \le x\sqrt{3} \}$
My attempt:
For $A$ I am using polar coordinates then $x^2+y^2=r^2$ and $\mathrm{d}x\mathrm{d}y=r\mathrm{d}r\mathrm{d}\phi$ and $x=r \cos(\phi)$ where $r$ goes from $0$ to $+\infty$ and $\phi$ from $0$ to $\pi$.
For $M$ I don't think I can use polar coordinates so I would let $0 \le y \le 1,\; 1 \le x \le \sqrt4,\; 0 \le z \le x\sqrt{3}$ and integrate $\mathrm{d}z\,\mathrm{d}x\,\mathrm{d}y$ in that order.
For the first one you did it right. Indeed, we have to evaluate: $$\int_{r=0}^{\infty}\frac{r^2}{1+r^2}dr\int_{\phi=0}^{\pi}\cos\phi d\phi$$ But for the second, you'd better do a simple plot for $x$ and $z$ as follows:
According to our assumptions, $y$ is independent of other variables $x$ and $z$ since, $$y|_0^1$$ Now for $x$ and $z$ (see the plot) we have: $$x|_{0.5}^1,~~z|_{\sqrt{1-x^2}}^{\sqrt{4-x^2}},~~~~~~~~~~~x|_1^2,~~z|_{0}^{\sqrt{4-x^2}}$$