Active Calculus - Multivariable

Subsection 11.8.1 Cylindrical Coordinates

Subsection 11.8.2 Triple Integrals in Cylindrical Coordinates

Activity 11.8.3.

A picture of a cylindrical box, \(B = \{(r,\theta,z) : r_1 \leq r \leq r_2, \theta_1 \leq \theta \leq \theta_2, z_1 \leq z \leq z_2\},\) is shown in Figure 11.8.2. 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 11.8.2.
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 11.8.2. 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?

Subsection 11.8.3 Spherical Coordinates

Subsection 11.8.4 Triple Integrals in Spherical Coordinates

Activity 11.8.6.

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 11.8.6. 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,

\begin{equation*} \Delta V \approx |PS| \ |\overset{\frown}{PR}| \ |\overset{\frown}{PQ}|, \end{equation*}

where \(|\overset{\frown}{PR}|\) 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 \(\overset{\frown}{PR}\text{?}\) (Hint: The arc \(\overset{\frown}{PR}\) 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 \(\overset{\frown}{PQ}\text{?}\) (Hint: The arc \(\overset{\frown}{PQ}\) lies on a horizontal circle as illustrated at right in Figure 11.8.6. 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.

Subsection 11.8.5 Summary

• The cylindrical coordinates of a point \(P\) are \((r,\theta,z)\) where \(r\) is the distance from the origin to the projection of \(P\) onto the \(xy\)-plane, \(\theta\) is the angle that the projection of \(P\) onto the \(xy\)-plane makes with the positive \(x\)-axis, and \(z\) is the vertical distance from \(P\) to the projection of \(P\) onto the \(xy\)-plane. When \(P\) has rectangular coordinates \((x,y,z)\text{,}\) it follows that its cylindrical coordinates are given by

  \begin{equation*} r^2 = x^2 + y^2, \ \ \ \ \ \tan(\theta) = \frac{y}{x}, \ \ \ \ \ z = z. \end{equation*}

  When \(P\) has given cylindrical coordinates \((r,\theta,z)\text{,}\) its rectangular coordinates are

  \begin{equation*} x = r \cos(\theta), \ \ \ \ \ y = r \sin(\theta), \ \ \ \ \ z = z. \end{equation*}
• The volume element \(dV\) in cylindrical coordinates is \(dV = r \, dz \, dr \, d\theta\text{.}\) Hence, a triple integral \(\iiint_S f(x,y,z) \, dA\) can be evaluated as the iterated integral

  \begin{equation*} \iiint_S f(r\cos(\theta), r\sin(\theta), z) \, r \, dz \, dr \, d\theta. \end{equation*}
• The spherical coordinates of a point \(P\) in 3-space are \(\rho\) (rho), \(\theta\text{,}\) and \(\phi\) (phi), where \(\rho\) is the distance from \(P\) to the origin, \(\theta\) is the angle that the projection of \(P\) onto the \(xy\)-plane makes with the positive \(x\)-axis, and \(\phi\) is the angle between the positive \(z\) axis and the vector from the origin to \(P\text{.}\) When \(P\) has Cartesian coordinates \((x,y,z)\text{,}\) the spherical coordinates are given by

  \begin{equation*} \rho^2 = x^2 + y^2 + z^2, \ \ \ \ \ \tan(\theta) = \frac{y}{x}, \ \ \ \ \ \cos(\phi) = \frac{z}{\rho}. \end{equation*}

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

  \begin{equation*} x = \rho \sin(\phi) \cos(\theta), \ \ \ \ \ y = \rho \sin(\phi) \sin(\theta), \ \ \ \ \ z = \rho \cos(\phi). \end{equation*}
• The volume element \(dV\) in spherical coordinates is \(dV = \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi\text{.}\) Thus, a triple integral \(\iiint_S f(x,y,z) \, dA\) can be evaluated as the iterated integral

  \begin{equation*} \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. \end{equation*}