Triple Integrals in Alternative Coordinate Systems: Learn It 1

  • Evaluate a triple integral by changing to cylindrical coordinates
  • Evaluate a triple integral by changing to spherical coordinates

Triple Integrals in Cylindrical and Spherical Coordinates

We have seen how to convert a double integral in rectangular coordinates into a double integral in polar coordinates in order to deal more conveniently with problems involving circular symmetry. A similar situation occurs with triple integrals, but here we need to distinguish between cylindrical symmetry and spherical symmetry. In this section we convert triple integrals in rectangular coordinates into a triple integral in either cylindrical or spherical coordinates.

The opera house l’Hemisphèric in Valencia, Spain, has four sections with one of the sections being a theater in a five-story-high sphere (ball) under an oval roof as long as a football field. Inside is an IMAX screen that changes the sphere into a planetarium with a sky full of [latex]9000[/latex] twinkling stars. Using triple integrals in spherical coordinates, we can find the volumes of different geometric shapes like these.

A photo of a glass ovoid dome reflecting in the water to make something like a football shape.
Figure 1. The City of Arts and Sciences in Valencia, Spain, has a unique structure along an axis of just two kilometers that was formerly the bed of the River Turia. The l’Hemisfèric has an IMAX cinema with three systems of modern digital projections onto a concave screen of 900 square meters. An oval roof over 100 meters long has been made to look like a huge human eye that comes alive and opens up to the world as the “Eye of Wisdom.” (credit: modification of work by Javier Yaya Tur, Wikimedia Commons)

Review of Cylindrical Coordinates

As we have seen earlier, in two-dimensional space [latex]\mathbb{R}^2[/latex], a point with rectangular coordinates [latex](x, y)[/latex] can be identified with [latex]{({r},{\theta})}[/latex] in polar coordinates and vice versa, where [latex]{x} = {r}{\cos}{\theta}[/latex], [latex]{y} = {r}{\sin}{\theta}[/latex], [latex]{r^2} = {x^2}+{y^2}[/latex], and [latex]{\tan}{\theta} = {({\frac{y}{x}})}[/latex] are the relationships between the variables.

In three-dimensional space [latex]\mathbb{R}^3[/latex], a point with rectangular coordinates [latex](x, y, z)[/latex] can be identified with cylindrical coordinates [latex]{({r},{\theta},{z})}[/latex] and vice versa. We can use these same conversion relationships, adding [latex]z[/latex] as the vertical distance to the point from the [latex]xy[/latex]-plane as shown in the following figure.

A box [latex]E[/latex] where the projection [latex]D[/latex] in the [latex]xy[/latex]-plane is of Type I.
Figure 2. Cylindrical coordinates are similar to polar coordinates with a vertical [latex]z[/latex] coordinate added.

From rectangular to cylindrical:

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

From cylindrical to rectangular:

  • [latex]{r^2} = {x^2} + {y^2}[/latex]
  • [latex]{\theta} = {\tan^{-1}}{({\frac{y}{x}})}[/latex] (note: you may need to add [latex]\pi[/latex] to reach the correct quadrant)
  • [latex]z = z[/latex]

In the two-dimensional plane with a rectangular coordinate system, when we say [latex]x=k[/latex] (constant) we mean an unbounded vertical line parallel to the [latex]y[/latex]-axis and when [latex]y=l[/latex] (constant) we mean an unbounded horizontal line parallel to the [latex]x[/latex]-axis. With the polar coordinate system, when we say [latex]r=c[/latex] (constant), we mean a circle of radius [latex]c[/latex] units and when [latex]{\theta} = {\alpha}[/latex] (constant) we mean an infinite ray making an angle [latex]{\alpha}[/latex] with the positive [latex]x[/latex]-axis.

Similarly, in three-dimensional space with rectangular coordinates [latex](x, y, z)[/latex], the equations [latex]x=k[/latex], [latex]y=l[/latex], and [latex]z=m[/latex], where [latex]k[/latex], [latex]l[/latex], and [latex]m[/latex] are constants, represent unbounded planes parallel to the [latex]yz[/latex]-plane, [latex]xz[/latex]-plane, and [latex]xy[/latex]-plane, respectively.

With cylindrical coordinates [latex]{({r},{\theta},{z})}[/latex], the equations [latex]{r} = {c}[/latex], [latex]{\theta} = {\alpha}[/latex], and [latex]z=m[/latex], where [latex]{c}[/latex], [latex]{\alpha}[/latex], and [latex]m[/latex] are constants, represent:

  • [latex]r = c[/latex]: an unbounded vertical cylinder with the [latex]z[/latex]-axis as its radial axis
  • [latex]\theta = \alpha[/latex]: a plane making a constant angle [latex]{\alpha}[/latex] with the [latex]xy[/latex]-plane
  • [latex]z = m[/latex]: an unbounded horizontal plane parallel to the [latex]xy[/latex]-plane

This means that the circular cylinder [latex]x^{2}+y^{2}=c^{2}[/latex] in rectangular coordinates can be represented simply as [latex]r=c[/latex] in cylindrical coordinates.

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.