<s>[INST] Compute the volume of the region $x^4 + y^2 + z^6 = 1.$ 
Compute the volume of the region $x^4 + y^2 + z^6 = 1.$

I can let $f(y) = 1-y^2 = x^4 + z^6.$ Then $z= \sqrt[6]{f(y)-x^4}.$ I suppose I can use symmetry to calculate the volume of one portion of the integral, but I'm not sure how to come up with the integral.
 [/INST] Technique due to Dirichlet. In the positive octant, volume is
$$ \frac{\Gamma \left( 1 + \frac{1}{2} \right)\Gamma \left( 1 + \frac{1}{4} \right)\Gamma \left( 1 + \frac{1}{6} \right)}{\Gamma \left(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{6} \right)}  $$
and the full volume is $8$ times that. If you can read the scans below, a crucial step is that
$$ \frac{1}{p} \; \Gamma \left( \frac{1}{p}   \right)  =  \Gamma \left( 1+ \frac{1}{p}   \right)    $$
For your problem, the volume in the positive octant is about $0.770182765,$ so the volume of the whole solid is about
$$ 6.1614621 $$
From gp-pari:
?  gamma( 3/2)*  gamma( 5/4) *  gamma( 7/6)  / gamma(23/12)
%6 = 0.7701827651926944255287920651
? 
? 8 *  gamma( 3/2)*  gamma( 5/4) *  gamma( 7/6)  / gamma(23/12)
%7 = 6.161462121541555404230336521
? 



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