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### Section 1-12 : 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.