from 2 equations we can get $$ 4x^2+y^2=8 $$ and it should be an ellipse which is $$ \frac{x^2}{2}+\frac{y^2}{8}=1. $$ I am stuck here because I can't find $r$. I searched online it involves $$ u^2+v^2=1. $$ I am confused and need some explanation .
2026-03-31 19:44:37.1774986277
Find the volume bounded by $4x^2+y^2=4z$ and $z=2$
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It looks like your ellipse is the values where the given subset of 3-space intersects the other subset (plane $z=2$). Though that's the right equation, it isn't what you need for the volume. Suggest you first find the range of possible $z$ values, and find the area of the "slice" at level $z$, then integrate that over your $z$ range. There's the formula $\pi a b$ for the area of an ellipse with major/minor axes $a,b$ which will be useful.
More details: For ease of notation let $h$ be a typical $z$ value between $0$ and $8.$ Then the cross section at level $z=h$ has equation $4x^2+y^2=h,$ which has major axis $\sqrt{h}$ and minor axis $\sqrt{h/2}.$ So the area $A(h)$ of the slice at level $h$ is $\pi a b = \pi h/\sqrt{2}.$ Integrate on $[0,8]$ and get (check it) $16 \pi \sqrt{2}$ for the volume.