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Section 6-2 : Parametric Surfaces

3. The cylinder \({x^2} + {y^2} = 5\) for \( - 1 \le z \le 6\).

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Because this surface is just a cylinder we just need the cylindrical coordinates conversion formulas with the polar coordinates in the \(xy\)-plane (since the cylinder is given in terms of \(x\) and \(y\)).

The conversion equations are,

\[x = r\cos \theta \hspace{0.5in}y = r\sin \theta \hspace{0.5in}z = z\]

However, recall that we are actually on the surface of the cylinder and so we know that \(r = \sqrt 5 \). The conversion equations are then,

\[x = \sqrt 5 \cos \theta \hspace{0.5in}y = \sqrt 5 \sin \theta \hspace{0.5in}z = z\] Show Step 2

We can now write down a set of parametric equations for the cylinder. They are,

\[\vec r\left( {z,\theta } \right) = \left\langle {x,y,z} \right\rangle = \left\langle {\sqrt 5 \cos \theta ,\sqrt 5 \sin \theta ,z} \right\rangle \]

Remember that all we do is plug the conversion formulas for \(x\), \(y\), and \(z\) into the \(x\), \(y\) and \(z\) components of the vector \(\left\langle {x,y,z} \right\rangle \) and we have a set of parametric equations. Also note that because the resulting vector equation is an equation in terms of \(z\) and \(\theta \) those will also be the variables for our set of parametric equation.

Show Step 3

Now, the only issue with the set of parametric equations above is that they are for the full cylinder and we don’t want that. We only want the cylinder in the given range of \(z\) so to finish this problem out all we need to do is add on a set of restrictions or ranges to our variables.

Doing that gives,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\vec r\left( {z,\theta } \right) = \left\langle {\sqrt 5 \cos \theta ,\sqrt 5 \sin \theta ,z} \right\rangle \,\,\,\,\,\,\,\,\, - 1 \le z \le 6\,\,\,,\,\,0 \le \theta \le 2\pi }}\]

Note that the \(z\) range is just the range given in the problem statement and the \(\theta \) range is the full zero to \(2\pi \) range since there was no mention of restricting the portion of the cylinder that we wanted with respect to \(\theta \) (for example, only the top half of the cylinder).