I am trying to calculate the volume of the solid from the following restrictions using double integral:
$$S=\{(x,y,z) \in \mathbb{R}^3:\ x^2+y^2 \le 4; 0 \le z \le x^2+y^2+4\}$$
From this I then graphed the following regions:
$$ x^2+y^2+z^2=16 \rightarrow \text{Ellipsoid} \\ x^2+y^2=4y \rightarrow \text{Paraboloid} $$
After graphing the regions and determining the desired section, I am not sure how to proceed especially in changing to polar coordinates.
Just change to cylindrical coordinates. Your domain of integration is a circle in the plane $z = 0$ of radius $2$. How can you write that in polar coordinates? And for $0 \leq z \leq x^2 +y^2 + 4$, what's this in polar coordinates? Once you've figured that out then recalling the Jacobian factor we have,
$$\textbf{Volume} = \int_{\theta = ?}^{\theta = ?} \int_{r = ?}^{r=?} r \int_{z = 0}^{z = ?} \ 1 \ dz \ dr \ d\theta $$