Define the two changes of coordinates needed

$T_1\=\left[x\=2 u\,y\=3 v\,z\=4 w\right]$$\stackrel{\text{assign}}{\to }$

$T_2\=\left[u\=\mathrm{\rho } \sin \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)\,v\=\mathrm{\rho } \sin \left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right)\,w\=\mathrm{\rho } \cos \left(\mathrm{\phi }\right)\right]$$\stackrel{\text{assign}}{\to }$

Make two successive changes of coordinates in the density function

$\genfrac{}{}{0ex}{}{\sqrt{1-{\left(x\/2\right)}^2-{\left(y\/3\right)}^2-{\left(z\/4\right)}^2}}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{T_1}$

$\frac{1}{12}\sqrt{-144u^2-144v^2-144w^2\+144}$

$\stackrel{\text{simplify}}{\=}$

$\sqrt{-u^2-v^2-w^2\+1}$

$\genfrac{}{}{0ex}{}{\sqrt{-u^2-v^2-w^2\+1}}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{T_2}$

$\sqrt{-{\mathrm{\ρ}}^2{\sin \left(\mathrm{\φ}\right)}^2{\cos \left(\mathrm{\θ}\right)}^2-{\mathrm{\ρ}}^2{\sin \left(\mathrm{\φ}\right)}^2{\sin \left(\mathrm{\θ}\right)}^2-{\mathrm{\ρ}}^2{\cos \left(\mathrm{\φ}\right)}^2\+1}$

$\stackrel{\text{simplify}}{\=}$

$\sqrt{-{\mathrm{\ρ}}^2\+1}$

Obtain the Jacobian of the transformation from $\left(x\,y\,z\right)$ to $\left(u\,v\,w\right)$ 

$\genfrac{}{}{0ex}{}{\left[x\,y\,z\right]}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{T_1}$ = $\left[2u\,3v\,4w\right]$$\stackrel{\text{Jacobian}}{\to }$ $\stackrel{\text{determinant}}{\to }$$24$$$

Integrate the density over the unit sphere

$24{\int }_0^{2 \mathrm{\π}}{\int }_0^{\mathrm{\π}}{\int }_0^1\sqrt{1-{\mathrm{\ρ}}^2} {\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right) \mathit{\ⅆ}\mathrm{\ρ} \mathit{\ⅆ}\mathrm{\φ} \mathit{\ⅆ}\mathrm{\θ}$ = $6{\mathrm{\π}}^2$$$

Tools≻Tasks≻Browse:
Calculus - Vector≻Integration≻Multiple Integration≻3-D≻Over a Sphere

Table 8.5.1(a)   Task template for integration over a sphere

Define the two changes of coordinates needed

$T_1≔\left[x\=2 u\,y\=3 v\,z\=4 w\right]\:$

$T_2≔\left[u\=\mathrm{\ρ} \sin \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)\,v\=\mathrm{\rho } \sin \left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right)\,w\=\mathrm{\rho } \cos \left(\mathrm{\phi }\right)\right]\:$

Make two successive changes of coordinates in the density function

$\mathrm{simplify}\left(\mathrm{eval}\left(\mathrm{eval}\left(\sqrt{1-{\left(x\/2\right)}^2-{\left(y\/3\right)}^2-{\left(z\/4\right)}^2}\,T_1\right)\,T_2\right)\right)$ = $\sqrt{-{\mathrm{\ρ}}^2\+1}$$$

Obtain the Jacobian of the transformation from $\left(x\,y\,z\right)$ to $\left(u\,v\,w\right)$

$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{Jacobian}\left(\left[2 u\,3 v\,4 w\right]\,\left[u\,v\,w\right]\,\mathrm{output}\=\mathrm{determinant}\right)$ = $24$$$

Integrate the density over the unit sphere

$$\begin{gathered} \mathrm{Int}\left(24\sqrt{1-{\mathrm{\ρ}}^2} {\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\,\left[\mathrm{\ρ}\=0..1\,\mathrm{\φ}\=0..\mathrm{\π}\,\mathrm{\θ}\=0..2 \mathrm{\π}\right]\right)\= \\ \mathrm{int}\left(24\sqrt{1-{\mathrm{\rho }}^2} {\mathrm{\rho }}^2\sin \left(\mathrm{\phi }\right)\,\left[\mathrm{\rho }\=0..1\,\mathrm{\phi }\=0..\mathrm{\pi }\,\mathrm{\theta }\=0..2 \mathrm{\pi }\right]\right) \end{gathered}$$

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{\mathrm{\π}}{\∫}_0^{1}24\sqrt{-{\mathrm{\ρ}}^2\+1}{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\ⅆ\mathrm{\ρ}\ⅆ\mathrm{\φ}\ⅆ\mathrm{\θ}\=6{\mathrm{\π}}^2$