\[ (x,y,z) = \left (\answer {2\cos (-\pi /4)\sin (\pi /4)}, \answer {2 \sin (-\pi /4)\sin (\pi /4)},\answer {2\cos (\pi /4)}\right ) \]
We integrate over regions in spherical coordinates.
Another way to generalize polar coordinates to three dimensions is with spherical coordinates.
An ordered triple consisting of a radius, an angle, and a height \((\rho ,\theta , \varphi )\) can be graphed as
![[Picture]](digInSphericalCoordinates.online-c1325e579e0394a8edbdcc60a5b608b3.svg)
Coordinates of this type are called spherical coordinates.
\begin{align*} x &= \rho \cdot \cos (\theta )\sin (\varphi )\\ y &= \rho \cdot \sin (\theta )\sin (\varphi )\\ z &= \rho \cdot \cos (\varphi ) \end{align*}
meaning:
Consider the point \((\rho ,\theta ,\phi )=(2,-\pi /4,\pi /4)\) in spherical coordinates. What is this point when expressed in
\((x,y,z)\)-coordinates?
1 Triple integrals in spherical 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 spherical coordinates and let’s express \(\d V\) in terms of \(\d \rho \), \(\d \phi \), and \(\d \theta \). To do this, consider the diagram below:
![[Picture]](digInSphericalCoordinates.online-7c64d26c3fbd4f986a14eafbeffa86d0.svg)
Here we see
\begin{align*} \d V &= (\rho \sin (\varphi )\d \theta )\cdot (\rho \d \varphi )\cdot (\d \rho )\\ &= \rho ^2 \sin (\varphi )\d \rho \d \varphi \d \theta . \end{align*}
Recalling that the determinate of a \(3\times 3\) matrix gives the volume of a parallelepiped, we
could also deduce the correct form for \(\d V\) by setting
\begin{align*} x(\rho ,\theta ,\phi ) &= \rho \cos (\theta )\sin (\phi )\\ y(\rho ,\theta ,\phi ) &= \rho \sin (\theta )\sin (\phi )\\ z(\rho ,\theta ,\phi ) &= \rho \cos (\phi ) \end{align*}
and computing:
\begin{align*} \d V &= \left | \det \begin{bmatrix} \pp [x]{\rho } \d \rho & \pp [y]{\rho } \d \rho & \pp [z]{\rho } \d \rho \\ \pp [x]{\theta } \d \theta & \pp [y]{\theta } \d \theta & \pp [z]{\theta } \d \theta \\ \pp [x]{\phi } \d \phi & \pp [y]{\phi } \d \phi & \pp [z]{\phi } \d \phi \end{bmatrix} \right |\\ &= \left | \det \begin{bmatrix} \answer [given]{\cos (\theta )\sin (\phi )} \d \rho & \answer [given]{\sin (\theta )\sin (\phi )} \d \rho & \answer [given]{\cos (\phi )} \d \rho \\ \answer [given]{-\rho \sin (\theta )\sin (\phi )} \d \theta & \answer [given]{\rho \cos (\theta )\sin (\phi )} \d \theta & 0\\ \answer [given]{\rho \cos (\theta )\cos (\phi )} \d \phi & \answer [given]{\rho \sin (\theta )\cos (\phi )} \d \phi & \answer [given]{-\rho \sin (\phi )}\d \phi \end{bmatrix} \right |\\ &= \rho ^2 \sin (\varphi )\d \rho \d \varphi \d \theta \end{align*}
We may now state at theorem:
Fubini Let \(F:\R ^3\to \R \) be continuous on the region
\[ R=\{(\rho ,\theta ,\phi ):\text {$\alpha \leq \theta \leq \beta $, $a\leq \phi \leq b$, $G_1(\theta ,\phi )\le \rho \le G_2(\theta ,\phi )$}\} \]
Then:
\begin{align*} \iiint _R &F(\rho ,\theta ,\phi )\d V\\ &= \int _\alpha ^\beta \int _{a}^{b} \int _{G_1(\theta ,\phi )}^{G_2(\theta ,\phi )}F(\rho ,\theta ,\phi ) \rho ^2 \sin (\phi )\d \rho \d \varphi \d \theta . \end{align*}
Write down a triple integral in spherical coordinates that will compute the volume
of a sphere of radius \(a\).
\[ \iiint _R \d V = \int _{\answer {0}}^{\answer {2\pi }} \int _{\answer {0}}^{\answer {\pi }} \int _{\answer {0}}^{\answer {a}} \rho ^2 \sin (\varphi )\d \rho \d \varphi \d \theta \]