5.5 Triple integrals in cylindrical and spherical coordinates  (Page 2/12)

Integration in cylindrical coordinates

Triple integrals can often be more readily evaluated by using cylindrical coordinates instead of rectangular coordinates. Some common equations of surfaces in rectangular coordinates along with corresponding equations in cylindrical coordinates are listed in [link] . These equations will become handy as we proceed with solving problems using triple integrals.

As before, we start with the simplest bounded region $B$ in ${ℝ}^3,$ to describe in cylindrical coordinates, in the form of a cylindrical box, $B=\left\{\left(r,\theta ,z\right)|a\le r\le b,\alpha \le \theta \le \beta ,c\le z\le d\right\}$ ( [link] ). Suppose we divide each interval into $l,m\text{and}n$ subdivisions such that $\text{Δ}r=\frac{b-a}{l},\text{Δ}\theta =\frac{\beta -\alpha }{m},$ and $\text{Δ}z=\frac{d-c}{n}.$ Then we can state the following definition for a triple integral in cylindrical coordinates.

Note that if $g\left(x,y,z\right)$ is the function in rectangular coordinates and the box $B$ is expressed in rectangular coordinates, then the triple integral ${\displaystyle \underset{B}{\iiint }g\left(x,y,z\right)dV}$ is equal to the triple integral ${\displaystyle \underset{B}{\iiint }g\left(r\text{cos}\theta ,r\text{sin}\theta ,z\right)rdrd\theta dz}$ and we have

As mentioned in the preceding section, all the properties of a double integral work well in triple integrals, whether in rectangular coordinates or cylindrical coordinates. They also hold for iterated integrals. To reiterate, in cylindrical coordinates, Fubini’s theorem takes the following form:

The iterated integral may be replaced equivalently by any one of the other five iterated integrals obtained by integrating with respect to the three variables in other orders.

Cylindrical coordinate systems work well for solids that are symmetric around an axis, such as cylinders and cones. Let us look at some examples before we define the triple integral in cylindrical coordinates on general cylindrical regions.