Section 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
References.
- Section 5.5 of OpenStax Calculus Volume 3.
1
openstax.org/books/calculus-volume-3/pages/5-5-triple-integrals-in-cylindrical-and-spherical-coordinates - Sections 15.7 and 15.8 of Calculus, Early Transcendentals by Stewart.
Subsection 5.5.1 Preview: Double Integrals in Polar Coordinates Revisited
To evaluate double integrals in cartesian coordinates \(x\text{,}\)\(y\) and in plane polar coordinates \(r\text{,}\)\(\theta\text{,}\) we use the iterated integral forms
\begin{equation*}
\iint\limits_D f \, dA = \iint\limits_D f(x,y)\, dx\, dy = \iint\limits_D f(r\cos \theta,r \sin \theta) r \, dr \, d\theta
\end{equation*}
The basic geometrical idea is embodied in the formula
\begin{equation}
dA = dx\, dy = r\, dr\, d\theta\tag{5.5.1}
\end{equation}
which intuitively speaking describes the area of an infinitesimal roughly rectangular region in terms of the infinitesimal ranges of the coordinates over that region.
Let us re-derive these formulas for the infinitesimal area \(dA\) in a heuristic, geometrical way, to prepare for the 3D versions.
For cartesian coordinates, it is clear that increasing \(x\) by up to \(dx\) and \(y\) by up to \(dy\) sweeps out a rectangle of sides \(dx\) and \(dy\) and thus of area \(dA = dx\, dy\text{.}\)
For plane polar coordinates, changing \(r\) by \(dr\) (without changing \(\theta\)) moves to a point a distance \(dr\) away, along a line through the origin, but instead changing the angle \(\theta\) by amount \(d\theta\) (without changing \(r\)) moves a distance \(r\, d\theta\) along a circle around the origin, so at right angles to the first movement.
Thus increasing \(r\) and \(\theta\) by amounts of up to \(dr\) and \(d\theta\) respectively sweeps out a roughly rectangular region of sides \(dr\) and \(r d\theta\text{,}\) and so of area \(dA = r\, dr\, d\theta\text{.}\)
Subsection 5.5.2 Triple Integrals in Cylindrical Coordinates
Recall cylindrical coordinates, introduced in Subsection 2.7.1, and in particular the change of coordinates formulas (2.7.1)
To express triple integrals in terms of three iterated integrals in these coordinates \(r\text{,}\) \(\theta\) and \(z\text{,}\) we need to describe the infinitesimal volume \(dV\) in terms of those coordinates and their differentials \(dr\text{,}\) \(d\theta\) and \(dx\text{.}\)
This is easy using the above results for plane polar coordinates, because varying the \(r\) and \(\theta\) coordinates by amounts of up to \(dr\) and \(d\theta\) again sweep out a horizontal area of \(r\ dr\ d\theta\text{,}\) and varying \(z\) by up to \(dz\) creates a "cylinder" of height \(dz\) over this region, thus having volume
\begin{equation}
dV = r\ dr\ d\theta\ dz.\tag{5.5.2}
\end{equation}
For a domain \(E\) of "Type \(dr\)-\(d\theta\)-\(dz\)" — meaning one described by inequalities \(a \leq z \leq b\text{,}\) \(f_1(z) \leq \theta \leq f_2(z)\text{,}\) \(g_1(\theta,z) \leq r \leq g_2(\theta,z)\) — the triple integral in cylindrical coordinates is given by "summing" over these pieces:
\begin{align}
\amp\iiint\limits_E f(x,y,z) dV =\notag\\
\amp\int_{z=a}^b\int_{\theta=f_1(z)}^{f_2(z)} \int_{r=g_1(\theta,z)}^{g_2(\theta,z)}
f(r \cos \theta, r \sin \theta, z)\, r\, dr\, d\theta\, dz.\tag{5.5.3}
\end{align}
Other domain shapes can be handled with other orders of the integrals. For the rest of this section details of domains and the limits of the three iterated integrals are omitted, and \(dV\) is loosely replaced by the form in Equation (5.5.2), or any of the five other versions got by reordering the differentials. For example,
\begin{equation*}
\iiint\limits_E f(x,y,z) dV = \iiint\limits_E f(r \cos \theta, r \sin \theta, z)\ dz\ rdr\ d\theta.
\end{equation*}
Subsection 5.5.3 Triple Integrals in Spherical Coordinates
Next we revisit spherical coordinates, introduced in Subsection 2.7.2, and in particular the change of coordinates formulas (2.7.4) and (2.7.5)
First, consider the volume of the small region near the point with spherical coordinates \((\rho,\phi,\theta)\) that is swept out by varying the coordinates by up to \(d\rho\text{,}\) \(d\phi\) and \(d\theta\) respectively.
- Changing \(\rho\) by \(d\rho\) moves a distance \(d\rho\) along a line through the origin, perpendicular to the sphere of radius \(\rho\text{.}\)
-
Changing \(\phi\) by \(d\phi\) moves on that sphere "southward" along a curve of longitude, of radius \(\rho\text{,}\) so the distance moved is \(\rho d\phi\text{.}\)This movement is perpendicular to the above movement due to changing \(\rho\text{.}\)
-
Changing \(\theta\) by \(d\theta\) moves on that sphere "eastward" along a circle of latitude of radius \(r = \rho \sin \phi\text{,}\) so the distance moved is \(r\, d\theta = \rho \sin \phi\, d\theta\text{.}\)This movement is perpendicular to both of the previous movements.
Altogether, changes in the three coordinates sweep our a region that is roughly a rectangular box of dimensions \(d\rho\) by \(\rho d\phi\) by \(\rho \sin \phi\, d\theta\text{,}\) and of infinitesimal volume
\begin{equation}
dV = d\rho\, (\rho\, d\phi)\, (\rho \sin \phi\, d\theta) = \rho^2 \sin\phi\ d\rho\ d\phi\ d\theta\tag{5.5.4}
\end{equation}
and so
\begin{align}
\amp\iiint\limits_E f(x,y,z)\ dV =\notag\\
\amp\iiint\limits_E f(\rho \sin\phi \cos \theta, \rho \sin\phi \sin \theta, \rho \cos\phi)
\rho^2 \sin\phi\ d\rho\ d\theta\ d\phi\tag{5.5.5}
\end{align}
Again there can be reordering of the three differentials to get various iterated integrals appropriate to various shapes of domain.
Study Guide.
Study Section 5.5 of OpenStax Calculus Volume 3; in particular
6
openstax.org/books/calculus-volume-3/pages/5-5-triple-integrals-in-cylindrical-and-spherical-coordinates- The two new forms of Fubini’s Theorem.
- All the Examples (and the Checkpoints following each).
- One or several exercises from each of the following ranges and pairs: 241–246, 249–252, 253–256, 267 & 268, 269–272.
