Question

Identify the surface whose equation is given.

\({\rm{2}}{{\rm{r}}^{\rm{2}}}{\rm{ + }}{{\rm{z}}^2}{\rm{ = 1}}\)

Short answer

The origin is the centre of the ellipsoid.

Step-by-step solution

Concept Introduction

A multiple integral is a definite integral of a function of many real variables in mathematics (particularly multivariable calculus), such as f(x, y) or f(x, y) (x, y, z). Integrals of a two-variable function over a region

Step 2:Explanation of the solution

Use the fact that \({{\rm{r}}^{\rm{2}}}{\rm{ = }}{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}\)once again.

After that, we may rewrite the equation as follows:

\({\rm{2}}{{\rm{x}}^{\rm{2}}}{\rm{ + 2}}{{\rm{y}}^{\rm{2}}}{\rm{ + }}{{\rm{z}}^{\rm{2}}}{\rm{ = 1}}\)

This is an ellipse with the origin at the centre and the following locations of interception:

\({\rm{x = y = \pm }}\frac{{\sqrt {\rm{2}} }}{{\rm{2}}}\)and \({\rm{z = 0}}\).

The origin is the centre of the ellipsoid.