5.5 Triple integrals in cylindrical and spherical coordinates  (Page 4/12)

Review of spherical coordinates

In three-dimensional space ${ℝ}^3$ in the spherical coordinate system, we specify a point $P$ by its distance $\rho$ from the origin, the polar angle $\theta$ from the positive $x\text{-axis}$ (same as in the cylindrical coordinate system), and the angle $\phi$ from the positive $z\text{-axis}$ and the line $OP$ ( [link] ). Note that $\rho \ge 0$ and $0\le \phi \le \pi .$ (Refer to Cylindrical and Spherical Coordinates for a review.) Spherical coordinates are useful for triple integrals over regions that are symmetric with respect to the origin.

Recall the relationships that connect rectangular coordinates with spherical coordinates.

From spherical coordinates to rectangular coordinates:

From rectangular coordinates to spherical coordinates:

Other relationships that are important to know for conversions are

and

The following figure shows a few solid regions that are convenient to express in spherical coordinates.

Integration in spherical coordinates

We now establish a triple integral in the spherical coordinate system, as we did before in the cylindrical coordinate system. Let the function $f\left(\rho ,\theta ,\phi \right)$ be continuous in a bounded spherical box, $B=\left\{\left(\rho ,\theta ,\phi \right)|a\le \rho \le b,\alpha \le \theta \le \beta ,\gamma \le \phi \le \psi \right\}.$ We then divide each interval into $l,m\text{and}n$ subdivisions such that $\text{Δ}\rho =\frac{b-a}{l},\text{Δ}\theta =\frac{\beta -\alpha }{m},\text{Δ}\phi =\frac{\psi -\gamma }{n}.$