We integrate over regions in cylindrical coordinates.

The first way we will generalize polar coordinates to three dimensions is with cylindrical coordinates.

Consider the point \((2, \pi /3,5)\) in cylindrical coordinates. What is this point when expressed in \((x,y,z)\)-coordinates?
\[ (x,y,z) = \left (\answer {2\cos (\pi /3)}, \answer {2 \sin (\pi /3)},\answer {5}\right ) \]
Consider the point \((1, -1,5)\) in \((x,y,z)\)-coordinates. What is this point when expressed in cylindrical coordinates where \(0\le \theta <2\pi \)?
\[ (r,\theta ,z) = \left (\answer {\sqrt {2}}, \answer {7\pi /4},\answer {5}\right ) \]

1 Triple integrals in cylindrical coordinates

If you want to evaluate this integral

\[ \iiint _R F \d V, \]

you have to change \(R\) to a region defined in \((x,y,z)\)-coordinates, and change \(\d V\) to some combination of \(\d x\d y\d z\) leaving you with some iterated integral:

\[ \int _a^b\int _c^d\int _p^q F(x,y,z) \d y \d x\d z \]

Now consider representing a region \(R\) in cylindrical coordinates and let’s express \(\d V\) in terms of \(\d r\), \(\d \theta \), and \(\d z\). To do this, consider the diagram below:

Here we see

\begin{align*} \d V &= \d r \cdot (r \d \theta )\cdot \d z \\ &= \d z\ r \d r \d \theta . \end{align*}
Write down a triple integral in cylindrical coordinates that will compute the volume of a cylinder of radius \(a\) and height \(h\).
\[ \iiint _R \d V = \int _{\answer {0}}^{\answer {2\pi }} \int _{\answer {0}}^{\answer {a}} \int _{\answer {0}}^{\answer {h}} \d z \ r \d r \d \theta \]