Find the volume of $\frac{x^2}{36}+\frac{y^2}{16}+\frac{z^2}{25}=1$ using linear transformation Is there anyone here that could help me calculating the volume of the ellipsoid $$\frac{x^2}{36}+\frac{y^2}{16}+\frac{z^2}{25}=1$$ by turning it into an circunference of coordinates $uvw$ using a linear transformation?
I have calculated this volume by integration and got $V=160\pi$.

Consider the transformation $(u,v,w) = f(x,y,z) = (\frac{x}{6}, \frac{y}{4}, \frac{z}{5})$
The given ellipsoid is transformed into $u^2+v^2+w^2=1$ which has volume $\frac{4\pi}{3}$. Now, the transformation also changes the volume from $V$ to $\frac{V}{6.4.5} = \frac{V}{120}$
So $\frac{V}{120} = \frac{4\pi}{3}$ so $V = 160\pi$
