Calculus III - Cylindrical Coordinates

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Section 12.12 : Cylindrical Coordinates

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6. Identify the surface generated by the equation : \({r^2} - 4r\cos \left( \theta \right) = 14\)

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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.

\[{x^2} + {y^2} - 4x = 14\] Show Step 2

To identify this equation (and you do know what it is!) let’s complete the square on the \(x\) part of the equation.

\[\begin{align*}{x^2} - 4x + {y^2} & = 14\\ {x^2} - 4x + 4 + {y^2} & = 14 + 4\\{\left( {x - 2} \right)^2} + {y^2} & = 18\end{align*}\]

So, we can see that this is a cylinder whose central axis is a vertical line parallel to the \(z\)-axis and goes through the point (2, 0) in the \(xy\)-plane and the radius of the cylinder is \(\sqrt {18} \).