Find the volume of the solid bounded by the surface
Find the volume of the solid bounded by the surface: $$ \left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)^n+\frac{z^{2n}}{c^{2n}}=\frac zh\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)^{n-1},\qquad n>1,\ a,b,c,h>0. $$
Explanation of Method
A note on this problem: unlike the other problems in this set, this one does not appear to admit a closed-form solution for general $n>1$, and is presented here as such rather than with a fabricated formula.
Writing $\rho=\frac{x^2}{a^2}+\frac{y^2}{b^2}$ (via standard elliptical polar coordinates, $\rho=r^2$), the defining equation becomes:
$$ \rho^n+\frac{z^{2n}}{c^{2n}}=\frac zh\rho^{n-1}. $$
For $n=1$ this is quadratic in $z$ and solvable in radicals: $\dfrac{z^2}{c^2}-\dfrac zh+\rho=0\Rightarrow z=\dfrac{c^2}{2h}\left(1\pm\sqrt{1-\frac{4h^2\rho}{c^4}}\right)$. But the problem explicitly restricts to $n>1$, and for $n\geqslant2$ the equation is a degree-$2n$ polynomial in $z$ (for fixed $\rho$) with coefficients depending on $\rho$, which does not solve in radicals for $n\geqslant3$ (by the Abel–Ruffini theorem) and does not yield an elementary closed-form height function $z(\rho)$ even for $n=2$. Consequently, the volume integral $V=2\pi ab\int_0^{\rho_{\max}}z(\rho)\,d\rho$ (or its polar equivalent) cannot be reduced to elementary functions for general $n>1$ without specifying a particular numerical value of $n$.
- Reduce to a single implicit equation in $\rho=\frac{x^2}{a^2}+\frac{y^2}{b^2}$ and $z$.
- Observe that this equation is only solvable in elementary closed form for $n=1$, which lies outside the stated domain $n>1$.
- Conclude that the volume, as posed for general $n>1$, requires either a specific numerical value of $n$ or numerical methods.
Step 1 · Reduce to a single implicit relation
Using $x=ar\cos\varphi,\,y=br\sin\varphi$ (Jacobian $abr$), $\rho=\frac{x^2}{a^2}+\frac{y^2}{b^2}=r^2$ depends only on $r$, so the surface equation becomes an implicit relation purely between $\rho$ and $z$:
$$ \rho^n+\frac{z^{2n}}{c^{2n}}=\frac zh\rho^{n-1}. $$
Step 2 · Why this resists a closed form for $n>1$
For a numeric target volume, one would need to solve this equation for $z$ as an explicit function of $\rho$, then integrate $z(\rho)\cdot\frac{ab}2$ (the elliptical-polar area element in $\rho$) over the relevant range of $\rho$. For $n=2$ this is already a quartic in $z$; for general integer $n$ it is a degree-$2n$ polynomial in $z$ whose coefficients depend on $\rho^n$ and $\rho^{n-1}$. Neither the polynomial's roots nor the resulting integral reduce to elementary functions for a symbolic $n$.
Step 3 · What can honestly be said
The only sub-case with an elementary closed form is $n=1$ (outside the problem's stated $n>1$ domain), where the surface is the sphere-like relation $\rho+\dfrac{z^2}{c^2}=\dfrac zh$, i.e. $\rho\leqslant\dfrac{c^4}{4h^2}$ with $z=\dfrac{c^2}{2h}\left(1\pm\sqrt{1-\frac{4h^2\rho}{c^4}}\right)$ — a lens-shaped solid between the two roots. For any specific numerical value of $n>1$ (e.g. $n=2,3,\ldots$), the volume could be evaluated numerically once $a,b,c,h$ are also given numbers, but no general symbolic closed form in $n$ exists.
Additional Notes
- A note on the source computation: the source's own attempt at this problem broke down in the same way — after setting up the same implicit relation between $\rho$ and $z$, it explicitly acknowledged "the general case requires numerical evaluation or special functions for arbitrary $n>1$," then nonetheless presented two different, mutually inconsistent, undefined-looking boxed expressions (one containing an unexplained "$I_n$," the other a specific formula $\dfrac{\pi abc^4}{12h^2}$ which is dimensionally and structurally inconsistent with the general-$n$ problem, and which does not correspond to the source's own worked $n=1$ special case either).
- Rather than present a fabricated closed form to fill this gap, this page states the honest mathematical obstruction: the defining equation is a degree-$2n$ polynomial in $z$ with $\rho$-dependent coefficients, which has no elementary closed-form solution for symbolic $n>1$. If a specific numerical $n$ (and numerical $a,b,c,h$) is intended, the volume is computable numerically on request, but the general symbolic answer requested by the problem as stated does not exist in elementary form.
Result
No general elementary closed form exists for $V(n)$ with $n>1$ symbolic; see the discussion above.