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.


1 Answer 1


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 $Z=1-X^2-Y^2$.


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