Volume from equation $(x ^2+ y ^2 + z ^2 ) ^2 = xyz$

Question

How can you calculate the volume of the shape represented by the following equation: $$(x ^2+ y ^2 + z ^2 ) ^2 = xyz$$ I tried converting it to polar form (so $r = \sin^2\left(\theta\right)\cdot\cos\left(\theta\right)\cdot\sin\left(\phi\right)\cdot\cos\left(\phi\right)$) and integrate over $\phi$ and $\theta$ with range $2\pi$, however, that didn't work out. How can I approach this?

Answer 1

$$ (x^2+y^2+z^2)^2 = xyz$$ $$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} r \cos \varphi \sin \psi \\ r \sin \varphi \sin \psi \\ r \cos \psi \end{pmatrix} $$ $$ r^4 =| r^3 \sin( \varphi) \cos( \varphi) \cos( \psi) \sin^2 (\psi) |$$ $$ r = |\sin( \varphi) \cos( \varphi) \cos( \psi) \sin^2 (\psi)| $$

$${\rm d}V = r^2 \sin(\psi) {\rm d}r {\rm d}\psi {\rm d}\varphi $$

$$ V = \iiint {\rm d}V = \int \limits_0^{2\pi} \int \limits_0^\pi \int_0^{|\sin( \varphi) \cos( \varphi) \cos( \psi) \sin^2 (\psi)|} r^2 \sin\psi\; {\rm d}r\,{\rm d}\psi\,{\rm d}\varphi $$

$$ V = \int \limits_0^{2\pi} \int \limits_0^\pi \frac{\sin\psi}{3} | \sin\varphi \cos\varphi \sin^2 \psi \cos \psi|^3\,{\rm d}\psi\,{\rm d}\varphi \\ = \int \limits_0^{2\pi} \frac{\sin^2\varphi \cos^2\varphi |\sin \varphi \cos \varphi|}{60}\;{\rm d}\varphi$$

According to Wolfram

$$ V = \frac{1}{180}$$