# Finding Volume Using Triple Integration

Set up and evaluate a triple integral to find the volume of the region bounded by the paraboloid $$z=1-\frac{x^2}{9}-\frac{y^2}{100}$$ and the $$xy$$-plane.

I understand I'm finding the volume of a paraboloid that forms a "dome" over the $$xy$$-plane. Moreover, I can see the paraboloid intersects with the $$xy$$-plane to form an ellipse given by $$\frac{x^2}{9}-\frac{y^2}{100}=1$$.

I have tried setting this up using rectangular coordinates but the integral started looking extremely messy. I then tried spherical coordinates but had trouble find the upper bound of $$\rho$$. Specifically, I can't seem to successfully translate the rectangular equation $$z=1-\frac{x^2}{9}-\frac{y^2}{100}$$ to a spherical equation and isolate $$\rho$$.

Any help would be appreciated.

Hint: If you do the change of variables$$\left\{\begin{array}{l}X=\frac x3\\Y=\frac y{10}\\Z=z,\end{array}\right.$$then you shall have to compute the volume under the paraboloid $$Z=1-X^2-Y^2$$.