Coordinate systems for triple integrals
3D Coordinates
Polar coordinates for the plane generalize to 3D in two different ways:
- Cylindrical coordinates
- Spherical coordinates
In this overview, we’ll meet both coordinate systems with a focus on expressing triple integrals
\[ \iiint_R f(x,y,z) \, dV \]
over regions that are “simple” in those coordinate systems.
Cylindrical coordinates
Cylindrical coordinates are like polar + \(z\): specify a point in the plane via polar coordinates \((r,\theta)\), then its vertical displacement using \(z\).
Cylindrical coords with xy
If you put \(x\) and \(y\) in that picture, you can see the relationship between cylindrical and Cartesian.
Translation
Given \(f(x,y,z)\), we can derive \(F(r,\theta,z)\) using
\[ \begin{aligned} x &= r\cos(\theta) \\ y &= r\sin(\theta) \\ z &= z \end{aligned} \]
That is,
\[ F(r,\theta,z) = f(r\cos(\theta), r\sin(\theta), z) \]
Examples
If \(f(x,y,z) = x^2 - \sin(yz)\), then
\[ F(r,\theta,z) = r^2\cos^2(\theta) - \sin(r\sin(\theta)z) \]
Any \(x^2 + y^2\) can be replaced with \(r^2\). For example, if
\(f(x,y,z) = (4 - (x^2 + y^2))z\), then
\[ F(r,\theta,z) = (4 - r^2)z \]
Triple integrals in cylindrical coordinates
We can express a triple integral as
\[ \color{green}{\iiint_R} \color{blue}{f(x,y,z)} \, \color{red}{dV} = \color{green}{\int_{\alpha}^{\beta} \int_c^d \int_a^b} \color{blue}{f(r\cos(\theta), r\sin(\theta), z)} \, \color{red}{r \, dr \, dz \, d\theta} \]
As with polar coordinates and general change of variables, we translate:
- bounds of integration
- the function
- the volume element
Note: the factor \(r\) is part of \(dV\).
Simple cylindrical regions
The simplest regions have the form
\[ a \leq r \leq b, \quad \alpha \leq \theta \leq \beta, \quad c \leq z \leq d \]
A functional bound
Sometimes bounds depend on functions. For example:
\[ 0 \leq r \leq 2, \quad 0 \leq \theta \leq 2\pi, \quad 0 \leq z \leq 4 - r^2 \]
Example
Set up
\[ \iiint_R (x^2 + y^2 + z)\, dV \]
in cylindrical coordinates for the region above.
Solution:
\[ \int_0^{2\pi} \int_0^2 \int_0^{4-r^2} (r^2 + z)\, r \, dz \, dr \, d\theta = \frac{64}{3}\pi \]
Spherical coordinates
- \(\rho\): distance to the origin
- \(\varphi\): angle from the positive \(z\)-axis
- \(\theta\): usual polar angle
Spherical to Cylindrical
Fix \(\theta\) to determine a plane, then relate \(\rho,\varphi\) to \(r,z\).
Spherical to Cylindrical (cont.)
Within that plane, the relationships follow from trigonometry.
Spherical to Cartesian
We can use cylindrical as an intermediate step:
\[ \begin{aligned} x &= r\cos(\theta) = \rho\sin(\varphi)\cos(\theta) \\ y &= r\sin(\theta) = \rho\sin(\varphi)\sin(\theta) \\ z &= \rho\cos(\varphi) \end{aligned} \]
Any \(x^2 + y^2 + z^2\) becomes \(\rho^2\).
Triple integrals in spherical coordinates
We can write
\[ \color{green}{\iiint_R} \color{blue}{f(x,y,z)} \, \color{red}{dV} = \color{green}{\int_{\alpha}^{\beta} \int_{\gamma}^{\delta} \int_a^b} \color{blue}{f(\rho\sin(\varphi)\cos(\theta), \rho\sin(\varphi)\sin(\theta), \rho\cos(\varphi))} \, \color{red}{\rho^2 \sin(\varphi)\, d\rho \, d\varphi \, d\theta} \]
Again, we translate:
- bounds
- function
- volume element
Now the factor \(\rho^2 \sin(\varphi)\) appears in \(dV\).
Simple spherical regions
These regions have the form
\[ \alpha \leq \theta \leq \beta, \quad \gamma \leq \varphi \leq \delta, \quad a \leq \rho \leq b \]
The volume element
Where does
\[ dV = \rho^2\sin(\varphi)\, d\rho\, d\varphi\, d\theta \]
come from?
From the Jacobian:
\[ \left| \begin{matrix} \partial x/\partial\rho & \partial x/\partial\varphi & \partial x/\partial\theta \\ \partial y/\partial\rho & \partial y/\partial\varphi & \partial y/\partial\theta \\ \partial z/\partial\rho & \partial z/\partial\varphi & \partial z/\partial\theta \end{matrix} \right| = \left| \begin{matrix} \sin(\varphi)\cos(\theta) & \sin(\varphi)\sin(\theta) & \cos(\varphi) \\ \rho\cos(\varphi)\cos(\theta) & \rho\cos(\varphi)\sin(\theta) & -\rho\sin(\varphi) \\ -\rho\sin(\varphi)\sin(\theta) & \rho\sin(\varphi)\cos(\theta) & 0 \end{matrix} \right| \]
Example
Compute the mass of a unit sphere with density
\[ f(x,y,z) = 2 - (x^2 + y^2 + z^2) \]
\[ \begin{aligned} \iiint_S (2-(x^2+y^2+z^2)) \, dV &= \int_0^{2\pi} \int_0^{\pi} \int_0^1 (2-\rho^2)\rho^2 \sin(\varphi)\, d\rho\, d\varphi\, d\theta \\ &= 2\pi \int_0^{\pi} \int_0^1 (2\rho^2 - \rho^4)\sin(\varphi)\, d\rho\, d\varphi \\ &= 2\pi \times 2 \left(\frac{2}{3} - \frac{1}{5}\right) \end{aligned} \]