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Section 12.12 : Cylindrical Coordinates

7. Identify the surface generated by the equation : \(z = 7 - 4{r^2}\)

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To identify the surface generated by this equation it’s probably best to first convert the equation into Cartesian coordinates. In this case that’s a pretty simple thing to do.

Here is the equation in Cartesian coordinates.

\[z = 7 - 4\left( {{x^2} + {y^2}} \right) = 7 - 4{x^2} - 4{y^2}\] Show Step 2

From the Cartesian equation in Step 1 we can see that the surface generated by the equation is an elliptic paraboloid that starts at \(z = 7\) and opens down.