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Section 12.12 : Cylindrical Coordinates

3. Convert the following equation written in Cartesian coordinates into an equation in Cylindrical coordinates.

\[{x^3} + 2{x^2} - 6z = 4 - 2{y^2}\]

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Start Solution

There really isn’t a whole lot to do here. All we need to do is plug in the following \(x\) and \(y\) polar conversion formulas into the equation where (and if) possible.

\[x = r\cos \theta \hspace{0.5in}y = r\sin \theta \hspace{0.5in}{r^2} = {x^2} + {y^2}\] Show Step 2

However, first we’ll do a little rewrite.

\[{x^3} + 2{x^2} + 2{y^2} - 6z = 4\hspace{0.5in} \to \hspace{0.5in}{x^3} + 2\left( {{x^2} + {y^2}} \right) - 6z = 4\] Show Step 3

Now let’s use the formulas from Step 1 to convert the equation into Cylindrical coordinates.

\[{\left( {r\cos \theta } \right)^3} + 2\left( {{r^2}} \right) - 6z = 4\hspace{0.5in} \to \hspace{0.5in}\require{bbox} \bbox[2pt,border:1px solid black]{{{r^3}{{\cos }^3}\theta + 2{r^2} - 6z = 4}}\]