Chapter 14.7, Problem 30ES

If a, b, and c are positive constants, then the transformation $x=au,y=b\upsilon ,z=cw$ can be rewritten as $x/a=u,y/b=\upsilon ,z/c=w,$ and hence it maps the spherical region $u^2+{\upsilon }^2+w^2\le 1$ into the ellipsoidal region $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\le 1$ In these exercises, perform the integration by transforming the ellipsoidal region of integration into a spherical region of integration and then evaluating the transformed integral in spherical coordinates.

${\displaystyle \underset{G}{\iiint }\left(y^2+z^2\right)dV,}$ where G is the region enclosed by the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$