Volume

$\text{Volume}\left[\text{ImplicitRegion}\left[z\geq 3 x^2+2 y^2\land 3 x^2+2 y^2+5 z^2\leq 1\land -0.35\leq x\leq 0.35\land -0.43\leq y\leq 0.43\land 0.\leq z\leq 0.45,\{x,y,z\}\right], \\ \text{Method}\to \text{Integrate}\right] \Longrightarrow 0.103495919745$ .

This is a ellipsoid cap over an elliptic paraboloid cup. By scaling the $x$ and $y$ coordinates to make the horizontal cross section circular, the integrations become much simpler. The volume can be unscaled back to the original dimensions afterwards linearly.

The cap: $\int_{\frac{1}{10} \left(\sqrt{21}-1\right)}^{\frac{1}{\sqrt{5}}} \left(1-5 z^2\right) \, dz \Longrightarrow -5 \left(\frac{1}{15 \sqrt{5}}-\frac{\left(\sqrt{21}-1\right)^3}{3000}\right)+\frac{1}{10} \left(1-\sqrt{21}\right)+\frac{1}{\sqrt{5}}$ .

The cup: $\int_0^{ \frac{1}{10} \left(\sqrt{21}-1\right)} q \, dq \Longrightarrow \frac{1}{200} \left(\sqrt{21}-1\right)^2$ .

Adding the cap and cup, multiplying by for the "circular" cross section and unscaling; $\frac{\pi \left(\frac{1}{200} \left(\sqrt{21}-1\right)^2-5 \left(\frac{1}{15 \sqrt{5}}-\frac{\left(\sqrt{21}-1\right)^3}{3000}\right)+\frac{1}{10} \left(1-\sqrt{21}\right)+\frac{1}{\sqrt{5}}\right)}{\sqrt{2} \sqrt{3}} \Longrightarrow \\ \frac{\left(40 \sqrt{5}-21 \sqrt{21}+31\right) \pi }{300 \sqrt{6}} \Longrightarrow \\ 0.103495912204$