Triple Integrals in Alternative Coordinate Systems: Learn It 3

Review of Spherical Coordinates

In three-dimensional space [latex]{\mathbb{R}^{3}}[/latex] in the spherical coordinate system, we specify a point [latex]P[/latex] by its distance [latex]{\rho}[/latex] from the origin, the polar angle [latex]{\theta}[/latex] from the positive [latex]x[/latex]-axis (same as in the cylindrical coordinate system), and the angle [latex]{\varphi}[/latex] from the positive [latex]z[/latex]-axis and the line [latex]OP[/latex] (Figure 1). Note that [latex]{\rho} \ {\geq} \ {0}[/latex] and [latex]{0} \ {\leq} \ {\varphi} \ {\leq} \ {\pi}[/latex].

When to Use Spherical Coordinates? Spherical coordinates are useful for triple integrals over regions that are symmetric with respect to the origin, such as spheres, cones, and other solids centered at the origin.
A depiction of the spherical coordinate system: a point (x, y, z) is shown, which is equal to (rho, theta, phi) in spherical coordinates. Rho serves as the spherical radius, theta serves as the angle from the x axis in the xy plane, and phi serves as the angle from the z axis.
Figure 1. The spherical coordinate system locates points with two angles and a distance from the origin.

Recall the relationships that connect rectangular coordinates with spherical coordinates.

From spherical to rectangular coordinates:

  • [latex]x = {\rho} \sin \varphi \cos \theta[/latex]
  • [latex]y = {\rho} \sin \varphi \sin \theta[/latex]
  • [latex]z = {\rho} \cos \varphi[/latex]

From rectangular to spherical coordinates:

  • [latex]{\rho}^{2} = {x^2} + {y^2} + {z^2}[/latex]
  • [latex]\tan\theta = \frac{y}{x}[/latex]
  • [latex]\varphi = \arccos\left ( \frac{z}{\sqrt{{x^2} + {y^2} + {z^2}}} \right )[/latex]

From spherical to cylindrical coordinates:

  • [latex]r = {\rho} \sin \varphi[/latex]
  • [latex]\theta = \theta[/latex]
  • [latex]z = {\rho} \cos \varphi[/latex]

From cylindrical to spherical coordinates:

  • [latex]\rho = \sqrt{{r^2}+{z^2}}[/latex]
  • [latex]\theta = \theta[/latex]
  • [latex]\varphi = \arccos\left ( \frac{z}{\sqrt{{r^2}+{z^2}}} \right )[/latex]

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

This figure consists of four figures. In the first, a sphere is shown with the note Sphere rho = c (constant). In the second, a half plane is drawn from the z axis with the note Half plane theta = c (constant). In the last two figures, a half cone is drawn in each with the note Half cone phi = c (constant). In the first of these, the cone opens up and it is marked 0 < c < pi/2. In the second of these, the cone opens down and it is marked pi/2 < c < pi.
Figure 2. Spherical coordinates are especially convenient for working with solids bounded by these types of surfaces. (The letter [latex]c[/latex] indicates a constant.)
For a more detailed review of cylindrical and spherical coordinates, including how to visualize these coordinate systems and work with them in three dimensions, refer to the section on Alternative Coordinate Systems and Surfaces.