Evaluating the Volume of a Cupola-Shaped Set by Integration

Let $f(x,y)=1-{\textstyle \frac{{x}^{2}}{4}}-{y}^{2}$ and $\mathrm{\Omega}=\{(x,y)\in {\mathbb{R}}^{2}:f(x,y)\ge 0\}.$

Compute the volume of the set $A=\{(x,y,z)\in {\mathbb{R}}^{3}:(x,y)\in \mathrm{\Omega},0\le z\le f(x,y)\}.$

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?

Let $f(x,y)=1-{\textstyle \frac{{x}^{2}}{4}}-{y}^{2}$ and $\mathrm{\Omega}=\{(x,y)\in {\mathbb{R}}^{2}:f(x,y)\ge 0\}.$

Compute the volume of the set $A=\{(x,y,z)\in {\mathbb{R}}^{3}:(x,y)\in \mathrm{\Omega},0\le z\le f(x,y)\}.$

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?