I'm working through the following example from the Princeton Review book:
If the ellipse $x^{2} + x^{2/9}=1$ in the $xz-$plane is revolved around the $z-$axis, what's the equation of the resulting ellipsoid surface?
Now, the curve $f(x,z)=x^{2}+x^{2/9}-1=0$ is being revolved around the $z-$axis, so I replace $x$ with $\pm \sqrt{x^{2}+y^{2}}$, and obtain $(\pm \sqrt{x^{2}+y^{2}})^{2}+(\pm \sqrt{x^{2}+y^{2}})^{2/9}=1$, which becomes $x^{2}+y^{2}+(x^{2}+y^{2})^{1/9}=1$.
My problem, however, is that the book gets $x^{2}+y^{2}+\frac{1}{9}z^{2}=1$. Which one of us is correct? And, if it's the book, how do they get $\frac{1}{9}z^{2}$ from $(\pm \sqrt{x^{2}+y^{2}})^{2/9}$?
Thank you in advance! :)
This can't be an ellipse, can it? For one thing, it's an equation with $x$ and nothing else. Clearly, a typo: one of two $x$s should be $z$. Even then, it's not an ellipse, unless we evict $/9$ from the exponent. So, the correct equation is $$x^{2} + \frac19 z^{2 }=1$$ Now if you do what you did, the result is what you expect.