11.8: Triple Integrals in Cylindrical and Spherical Coordinates

Motivating Questions

We have encountered two different coordinate systems in \(\mathbb^2\) — the rectangular and polar coordinates systems — and seen how in certain situations, polar coordinates form a convenient alternative. In a similar way, there are two additional natural coordinate systems in \(\mathbb^3\text<.>\) Given that we are already familiar with the Cartesian coordinate system for \(\mathbb^3\text\) we next investigate the cylindrical and spherical coordinate systems (each of which builds upon polar coordinates in \(\mathbb^2\)). In what follows, we will see how to convert among the different coordinate systems, how to evaluate triple integrals using them, and some situations in which these other coordinate systems prove advantageous.

Preview Activity \(\PageIndex\)

In the following questions, we investigate the two new coordinate systems that are the subject of this section: cylindrical and spherical coordinates. Our goal is to consider some examples of how to convert from rectangular coordinates to each of these systems, and vice versa. Triangles and trigonometry prove to be particularly important.

  1. The cylindrical coordinates of a point in \(\mathbb^3\) are given by \((r,\theta,z)\) where \(r\) and \(\theta\) are the polar coordinates of the point \((x, y)\) and \(z\) is the same \(z\) coordinate as in Cartesian coordinates. An illustration is given at left in Figure \(\PageIndex\).
    1. Find cylindrical coordinates for the point whose Cartesian coordinates are \((-1, \sqrt, 3)\text<.>\) Draw a labeled picture illustrating all of the coordinates.
    2. Find the Cartesian coordinates of the point whose cylindrical coordinates are \(\left(2, \frac<5\pi>, 1\right)\text<.>\) Draw a labeled picture illustrating all of the coordinates.
  2. The spherical coordinates of a point in \(\mathbb^3\) are \(\rho\) (rho), \(\theta\text\) and \(\phi\) (phi), where \(\rho\) is the distance from the point to the origin, \(\theta\) has the same interpretation it does in polar coordinates, and \(\phi\) is the angle between the positive \(z\) axis and the vector from the origin to the point, as illustrated at right in Figure \(\PageIndex\). You should convince yourself that any point in \(\mathbb^3\) can be represented in spherical coordinates with \(\rho \geq 0\text\) \(0 \leq \theta \lt 2 \pi\text\) and \(0 \leq \phi \leq \pi\text<.>\) For the following questions, consider the point \(P\) whose Cartesian coordinates are \((-2,2,\sqrt)\text<.>\)
    1. What is the distance from \(P\) to the origin? Your result is the value of \(\rho\) in the spherical coordinates of \(P\text<.>\)
    2. Determine the point that is the projection of \(P\) onto the \(xy\)-plane. Then, use this projection to find the value of \(\theta\) in the polar coordinates of the projection of \(P\) that lies in the plane. Your result is also the value of \(\theta\) for the spherical coordinates of the point.
    3. Based on the illustration in Figure \(\PageIndex\), how is the angle \(\phi\) determined by \(\rho\) and the \(z\) coordinate of \(P\text\) Use a well-chosen right triangle to find the value of \(\phi\text\) which is the final component in the spherical coordinates of \(P\text<.>\) Draw a carefully labeled picture that clearly illustrates the values of \(\rho\text\) \(\theta\text\) and \(\phi\) in this example, along with the original rectangular coordinates of \(P\text<.>\)
    4. Based on your responses to i., ii., and iii., if we are given the Cartesian coordinates \((x,y,z)\) of a point \(Q\text\) how are the values of \(\rho\text\) \(\theta\text\) and \(\phi\) in the spherical coordinates of \(Q\) determined by \(x\text\) \(y\text\) and \(z\text\)

Cylindrical Coordinates

As we stated in Preview Activity \(\PageIndex\), the cylindrical coordinates of a point are \((r,\theta,z)\text\) where \(r\) and \(\theta\) are the polar coordinates of the point \((x, y)\text\) and \(z\) is the same \(z\) coordinate as in Cartesian coordinates. The general situation is illustrated Figure \(\PageIndex\).

Since we already know how to convert between rectangular and polar coordinates in the plane, and the \(z\) coordinate is identical in both Cartesian and cylindrical coordinates, the conversion equations between the two systems in \(\mathbb^3\) are essentially those we found for polar coordinates.

Converting between Cartesian and cylindrical coordinates
\[ r^2 = x^2 + y^2 \ \ \ \ \ \tan(\theta) = \frac \ \ \ \ \ \text < and >\ \ \ \ \ z = z, \nonumber \]

Just as with rectangular coordinates, where we usually write \(z\) as a function of \(x\) and \(y\) to plot the resulting surface, in cylindrical coordinates, we often express \(z\) as a function of \(r\) and \(\theta\text<.>\) In the following activity, we explore several basic equations in cylindrical coordinates and the corresponding surface each generates.

Activity \(\PageIndex\)

In this activity, we graph some surfaces using cylindrical coordinates. To improve your intuition and test your understanding, you should first think about what each graph should look like before you plot it using appropriate technology.

  1. What familiar surface is described by the points in cylindrical coordinates with \(r=2\text\) \(0 \leq \theta \leq 2\pi\text\) and \(0 \leq z \leq 2\text\) How does this example suggest that we call these coordinates cylindrical coordinates? How does your answer change if we restrict \(\theta\) to \(0 \leq \theta \leq \pi\text\)
  2. What familiar surface is described by the points in cylindrical coordinates with \(\theta=2\text\) \(0 \leq r \leq 2\text\) and \(0 \leq z \leq 2\text\)
  3. What familiar surface is described by the points in cylindrical coordinates with \(z=2\text\) \(0 \leq \theta \leq 2\pi\text\) and \(0 \leq r \leq 2\text\)
  4. Plot the graph of the cylindrical equation \(z=r\text\) where \(0 \leq \theta \leq 2\pi\) and \(0 \leq r \leq 2\text\) What familiar surface results?
  5. Plot the graph of the cylindrical equation \(z= \theta\) for \(0 \leq \theta \leq 4 \pi\text\) What does this surface look like?

As the name and Activity \(\PageIndex\) suggests, cylindrical coordinates are useful for describing surfaces that are cylindrical in nature.

Triple Integrals in Cylindrical Coordinates

To evaluate a triple integral \(\iiint_S f(x,y,z) \, dV\) as an iterated integral in Cartesian coordinates, we use the fact that the volume element \(dV\) is equal to \(dz \, dy \, dx\) (which corresponds to the volume of a small box). To evaluate a triple integral in cylindrical coordinates, we similarly must understand the volume element \(dV\) in cylindrical coordinates.

Activity \(\PageIndex\)

A picture of a cylindrical box, \(B = \,\) is shown in Figure \(\PageIndex\). Let \(\Delta r = r_2-r_1\text\) \(\Delta \theta = \theta_2 - \theta_1\text\) and \(\Delta z = z_2-z_1\text<.>\) We want to determine the volume \(\Delta V\) of \(B\) in terms of \(\Delta r\text\) \(\Delta \theta\text\) \(\Delta z\text\) \(r\text\) \(\theta\text\) and \(z\text<.>\)

  1. Appropriately label \(\Delta r\text\) \(\Delta \theta\text\) and \(\Delta z\) in Figure \(\PageIndex\).
  2. Let \(\Delta A\) be the area of the projection of the box, \(B\text\) onto the \(xy\)-plane, which is shaded blue in Figure \(\PageIndex\). Recall that we previously determined the area \(\Delta A\) in polar coordinates in terms of \(r\text\) \(\Delta r\text\) and \(\Delta \theta\text<.>\) In light of the fact that we know \(\Delta A\) and that \(z\) is the standard \(z\) coordinate from Cartesian coordinates, what is the volume \(\Delta V\) in cylindrical coordinates?

Activity \(\PageIndex\) demonstrates that the volume element \(dV\) in cylindrical coordinates is given by \(dV = r \, dz \, dr \, d\theta\text\) and hence the following rule holds in general.

Triple integrals in cylindrical coordinates

Given a continuous function \(f = f(x,y,z)\) over a region \(S\) in \(\mathbb^3\text\)

\[ \iiint_S f(x,y,z) \, dV = \iiint_S f(r\cos(\theta), r\sin(\theta), z) \, r \, dz \, dr \, d\theta. \nonumber \]

The latter expression is an iterated integral in cylindrical coordinates.

Of course, to complete the task of writing an iterated integral in cylindrical coordinates, we need to determine the limits on the three integrals: \(\theta\text\) \(r\text\) and \(z\text<.>\) In the following activity, we explore how to do this in several situations where cylindrical coordinates are natural and advantageous.

Activity \(\PageIndex\)

In this activity we work with triple integrals in cylindrical coordinates.

fig_11_8_two_cones.svg

  1. Let \(S\) be the solid bounded above by the graph of \(z = x^2+y^2\) and below by \(z=0\) on the unit disk in the \(xy\)-plane.
    1. The projection of the solid \(S\) onto the \(xy\)-plane is a disk. Describe this disk using polar coordinates.
    2. Now describe the surfaces bounding the solid \(S\) using cylindrical coordinates.
    3. Determine an iterated triple integral expression in cylindrical coordinates that gives the volume of \(S\text<.>\) You do not need to evaluate this integral.
  2. Suppose the density of the cone defined by \(r = 1 - z\text\) with \(z \geq 0\text\) is given by \(\delta(r, \theta, z) = z\text<.>\) A picture of the cone is shown at left in Figure \(\PageIndex\), and the projection of the cone onto the \(xy\)-plane in given at right in Figure \(\PageIndex\). Set up an iterated integral in cylindrical coordinates that gives the mass of the cone. You do not need to evaluate this integral.
  3. Determine an iterated integral expression in cylindrical coordinates whose value is the volume of the solid bounded below by the cone \(z = \sqrt\) and above by the cone \(z = 4 - \sqrt\text<.>\) A picture is shown in Figure \(\PageIndex\). You do not need to evaluate this integral.

Spherical Coordinates

As we saw in Preview Activity \(\PageIndex\), the spherical coordinates of a point in 3-space have the form \((\rho, \theta, \phi)\text\) where \(\rho\) is the distance from the point to the origin, \(\theta\) has the same meaning as in polar coordinates, and \(\phi\) is the angle between the positive \(z\) axis and the vector from the origin to the point. The overall situation is illustrated at right in Figure \(\PageIndex\).

The example in Preview Activity \(\PageIndex\) and Figure \(\PageIndex\) suggest how to convert between Cartesian and spherical coordinates.

Coverting between Cartesian and spherical coordinates

\[ \rho = \sqrt \ \ \ \ \ \tan(\theta) = \frac \ \ \ \ \ \text < and >\ \ \ \ \ \cos(\phi) = \frac, \nonumber \]

\[ x = \rho \sin(\phi) \cos(\theta) \ \ \ \ \ y = \rho \sin(\phi) \sin(\theta) \ \ \ \ \ \text < and >\ \ \ \ \ z = \rho \cos(\phi). \nonumber \]

When it comes to thinking about particular surfaces in spherical coordinates, similar to our work with cylindrical and Cartesian coordinates, we usually write \(\rho\) as a function of \(\theta\) and \(\phi\text\) this is a natural analog to polar coordinates, where we often think of our distance from the origin in the plane as being a function of \(\theta\text<.>\) In spherical coordinates, we likewise often view \(\rho\) as a function of \(\theta\) and \(\phi\text\) thus viewing distance from the origin as a function of two key angles.

In the following activity, we explore several basic equations in spherical coordinates and the surfaces they generate.

Activity \(\PageIndex\)

In this activity, we graph some surfaces using spherical coordinates. To improve your intuition and test your understanding, you should first think about what each graph should look like before you plot it using appropriate technology.

  1. What familiar surface is described by the points in spherical coordinates with \(\rho = 1\text\) \(0 \leq \theta \leq 2\pi\text\) and \(0 \leq \phi \leq \pi\text\) How does this particular example demonstrate the reason for the name of this coordinate system? What if we restrict \(\phi\) to \(0 \leq \phi \leq \frac<\pi>\text\)
  2. What familiar surface is described by the points in spherical coordinates with \(\phi = \frac<\pi>\text\) \(0 \leq \rho \leq 1\text\) and \(0 \leq \theta \leq 2\pi\text\)
  3. What familiar shape is described by the points in spherical coordinates with \(\theta = \frac<\pi>\text\) \(0 \leq \rho \leq 1\text\) and \(0 \leq \phi \leq \pi\text\)
  4. Plot the graph of \(\rho = \theta\text\) for \(0 \leq \phi \leq \pi\) and \(0 \leq \theta \leq 2 \pi\text\) How does the resulting surface appear?

As the name and Activity \(\PageIndex\) indicate, spherical coordinates are particularly useful for describing surfaces that are spherical in nature; they are also convenient for working with certain conical surfaces.

Triple Integrals in Spherical Coordinates

As with rectangular and cylindrical coordinates, a triple integral \(\iiint_S f(x,y,z) \, dV\) in spherical coordinates can be evaluated as an iterated integral once we understand the volume element \(dV\text<.>\)

Activity \(\PageIndex\)

To find the volume element \(dV\) in spherical coordinates, we need to understand how to determine the volume of a spherical box of the form \(\rho_1 \leq \rho \leq \rho_2\) (with \(\Delta \rho = \rho_2-\rho_1)\text\) \(\phi_1 \leq \phi \leq \phi_2\) (with \(\Delta \phi = \phi_2-\phi_1\)), and \(\theta_1 \leq \theta \leq \theta_2\) (with \(\Delta \theta = \theta_2-\theta_1\)). An illustration of such a box is given at left in Figure \(\PageIndex\). This spherical box is a bit more complicated than the cylindrical box we encountered earlier. In this situation, it is easier to approximate the volume \(\Delta V\) than to compute it directly. Here we can approximate the volume \(\Delta V\) of this spherical box with the volume of a Cartesian box whose sides have the lengths of the sides of this spherical box. In other words,

\[ \Delta V \approx |PS| \ |\overset| \ |\overset|, \nonumber \]

where \(|\overset|\) denotes the length of the circular arc from \(P\) to \(R\text<.>\)

  1. What is the length \(|PS|\) in terms of \(\rho\text\)
  2. What is the length of the arc \(\overset\text\) (Hint: The arc \(\overset\) is an arc of a circle of radius \(\rho_2\text\) and arc length along a circle is the product of the angle measure (in radians) and the circle's radius.)
  3. What is the length of the arc \(\overset\text\) (Hint: The arc \(\overset\) lies on a horizontal circle as illustrated at right in Figure \(\PageIndex\). What is the radius of this circle?)
  4. Use your work in (a), (b), and (c) to determine an approximation for \(\Delta V\) in spherical coordinates.

Letting \(\Delta \rho\text\) \(\Delta \theta\text\) and \(\Delta \phi\) go to 0, it follows from the final result in Activity \(\PageIndex\) that \(dV = \rho^2 \, \sin(\phi) \, d\rho \, d\theta \, d\phi \) in spherical coordinates, and thus allows us to state the following general rule.

Triple integrals in spherical coordinates

Given a continuous function \(f = f(x,y,z)\) over a region \(S\) in \(\mathbb^3\text\) the triple integral \(\iiint_S f(x,y,z) \, dV\) is converted to the integral

\[ \iiint_S f(\rho\sin(\phi)\cos(\theta), \rho \sin(\phi) \sin(\theta), \rho \cos(\phi)) \, \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi \nonumber \]

in spherical coordinates.

The latter expression is an iterated integral in spherical coordinates.

Finally, in order to actually evaluate an iterated integral in spherical coordinates, we must of course determine the limits of integration in \(\phi\text\) \(\theta\text\) and \(\rho\text<.>\) The process is similar to our earlier work in the other two coordinate systems.

Activity \(\PageIndex\)

We can use spherical coordinates to help us more easily understand some natural geometric objects.

  1. Recall that the sphere of radius \(a\) has spherical equation \(\rho = a\text\) Set up and evaluate an iterated integral in spherical coordinates to determine the volume of a sphere of radius \(a\text\)
  2. Set up, but do not evaluate, an iterated integral expression in spherical coordinates whose value is the mass of the solid obtained by removing the cone \(\phi=\frac<\pi>\) from the sphere \(\rho = 2\) if the density \(\delta\) at the point \((x,y,z)\) is \(\delta(x,y,z) = \sqrt\text\) An illustration of the solid is shown in Figure \(\PageIndex\).

Summary

\[ r^2 = x^2 + y^2, \ \ \ \ \ \tan(\theta) = \frac, \ \ \ \ \ z = z. \nonumber \] When \(P\) has given cylindrical coordinates \((r,\theta,z)\text<,>\) its rectangular coordinates are \[ x = r \cos(\theta), \ \ \ \ \ y = r \sin(\theta), \ \ \ \ \ z = z. \nonumber \] \[ \iiint_S f(r\cos(\theta), r\sin(\theta), z) \, r \, dz \, dr \, d\theta. \nonumber \]

\[ \rho^2 = x^2 + y^2 + z^2, \ \ \ \ \ \tan(\theta) = \frac, \ \ \ \ \ \cos(\phi) = \frac. \nonumber \]

Given the point \(P\) in spherical coordinates \((\rho, \phi, \theta)\text<,>\) its rectangular coordinates are

\[ x = \rho \sin(\phi) \cos(\theta), \ \ \ \ \ y = \rho \sin(\phi) \sin(\theta), \ \ \ \ \ z = \rho \cos(\phi). \nonumber \]

\[ \iiint_S f(\rho\sin(\phi)\cos(\theta), \rho \sin(\phi) \sin(\theta), \rho \cos(\phi)) \, \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi. \nonumber \]

This page titled 11.8: Triple Integrals in Cylindrical and Spherical Coordinates is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Matthew Boelkins, David Austin & Steven Schlicker (ScholarWorks @Grand Valley State University) via source content that was edited to the style and standards of the LibreTexts platform.

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