Solve integral $\iiint_Ax^pdxdydz$ on $A=\{(x,y,z):x^2+y^2+z^2We have following integral to count:
$$
\iiint_Ax^pdxdydz
$$
where $p$ is constant real number and $A=\{(x,y,z):x^2+y^2+z^2<x^{\frac{1}{3}}\}$
I tried spherical substitution and cylindrical.

By setting $x=w^3$, the triple integral turns into
$$3\iiint_{w^6+y^2+z^2<w}w^{3p+2}\,dw\,dy\,dz \tag{1}$$
that equals:
$$ 3 \int_{0}^{1}\iint_{y^2+z^2<w-w^6}w^{3p+2}\,dy\,dz   \,dw\tag{2}$$
or:
$$ 3\pi \int_{0}^{1}w^{3p+2}(w-w^6)\,dw = \color{red}{\frac{5\pi}{(p+3)(3p+4)}}.\tag{3}$$
