Let $f(x,y) = 1 - \tfrac{x^2}{4} - y^2$ and $$ \Omega = \lbrace (x,y) \in \mathbb{R}^2 \colon f(x,y) \ge 0 \rbrace. $$ Compute the volume of the set $$ A = \lbrace (x,y,z) \in \mathbb{R}^3 \colon (x,y) \in \Omega, 0 \le z \le f(x,y) \rbrace. $$
My idea is to slice the set along the $z$-axis, obtaining a set $E_z$ - in fact, an ellipse - and computing the volume as $$ \int_0^1 \int_{E_z} dxdydz $$
However, I am stuck finding a way to describe $E_z$. What is the best strategy to do that?
Use this parameterization for the whole $3$ dimensional space
$$\begin{align} x&=2u\cos v \\ y&=u \sin v \\ z&=w \end{align} \qquad 0 \le u \lt \infty, \quad 0 \le v \lt 2\pi, \quad -\infty \lt w \lt \infty$$
then the integral for volume will be
$$V=\int_{v=0}^{2\pi}\int_{u=0}^{1}\int_{w=0}^{1-u^2}\frac{\partial(x,y,z)}{\partial(u,v,w)}dw\,du\,dv$$
where
$$\frac{\partial(x,y,z)}{\partial(u,v,w)}= \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{vmatrix}= \begin{vmatrix} 2\cos v & -2u \sin v & 0 \\ \sin v & u \cos v & 0 \\ 0 & 0 & 1 \end{vmatrix}= 2u$$
is the jacobian.