∞Calculus IV Unit 15 – Triple Integrals in Spherical Coordinates

What's the Big Idea?

• Triple integrals in spherical coordinates enable the calculation of volume, mass, and other physical quantities for three-dimensional objects with spherical symmetry
• Spherical coordinates $(r, \theta, \phi)$ are a natural choice for objects with spherical or nearly spherical geometry, such as spheres, ellipsoids, and regions bounded by spherical surfaces
• The transformation from Cartesian coordinates $(x, y, z)$ to spherical coordinates involves trigonometric functions and introduces a Jacobian determinant in the integral
• Spherical coordinates simplify the integration process by aligning the coordinate system with the object's symmetry, reducing the complexity of the integral limits and integrand
• Triple integrals in spherical coordinates have applications in physics, engineering, and mathematics, including calculating moments of inertia, electric fields, and gravitational potentials

Key Concepts

• Spherical coordinates $(r, \theta, \phi)$:
  • $r$: radial distance from the origin
  • $\theta$: polar angle measured from the positive $z$-axis $(0 \leq \theta \leq \pi)$
  • $\phi$: azimuthal angle measured in the $xy$-plane from the positive $x$-axis $(0 \leq \phi < 2\pi)$
• Coordinate transformation:
  • $x = r \sin\theta \cos\phi$
  • $y = r \sin\theta \sin\phi$
  • $z = r \cos\theta$
• Jacobian determinant in spherical coordinates: $r^2 \sin\theta$
• Volume element in spherical coordinates: $dV = r^2 \sin\theta \, dr \, d\theta \, d\phi$
• Spherical shells: regions bounded by two concentric spheres with radii $r_1$ and $r_2$
• Spherical sectors: regions bounded by two meridional planes $(\phi_1, \phi_2)$ and a cone $(\theta_1, \theta_2)$
• Symmetry in spherical coordinates: objects with rotational symmetry about the $z$-axis or reflection symmetry about the $xy$-plane can lead to simplified integral limits

Coordinate System Breakdown

• Spherical coordinates define a point in 3D space using a radial distance $r$, a polar angle $\theta$, and an azimuthal angle $\phi$
• The radial distance $r$ represents the distance from the origin to the point, with $r \geq 0$
• The polar angle $\theta$ is measured from the positive $z$-axis, with $0 \leq \theta \leq \pi$
  • $\theta = 0$ corresponds to the positive $z$-axis
  • $\theta = \pi/2$ corresponds to the $xy$-plane
  • $\theta = \pi$ corresponds to the negative $z$-axis
• The azimuthal angle $\phi$ is measured in the $xy$-plane from the positive $x$-axis, with $0 \leq \phi < 2\pi$
  • $\phi = 0$ corresponds to the positive $x$-axis
  • $\phi = \pi/2$ corresponds to the positive $y$-axis
• The transformation from Cartesian to spherical coordinates is given by:
  • $x = r \sin\theta \cos\phi$
  • $y = r \sin\theta \sin\phi$
  • $z = r \cos\theta$
• The inverse transformation from spherical to Cartesian coordinates is:
  • $r = \sqrt{x^2 + y^2 + z^2}$
  • $\theta = \arccos\left(\frac{z}{\sqrt{x^2 + y^2 + z^2}}\right)$
  • $\phi = \arctan2(y, x)$, where $\arctan2$ is the two-argument arctangent function

Setting Up Triple Integrals

• To set up a triple integral in spherical coordinates, follow these steps:
  1. Identify the region of integration and its boundaries in terms of $r$, $\theta$, and $\phi$
  2. Determine the order of integration (usually $dr$, $d\theta$, $d\phi$, but it may vary depending on the region)
  3. Write the integral with the appropriate limits of integration and the Jacobian determinant $r^2 \sin\theta$
  4. Express the integrand in terms of spherical coordinates $(r, \theta, \phi)$ if necessary
• The general form of a triple integral in spherical coordinates is: $\iiint_E f(r, \theta, \phi) \, r^2 \sin\theta \, dr \, d\theta \, d\phi$ where $E$ is the region of integration and $f(r, \theta, \phi)$ is the integrand
• When setting up the limits of integration, consider the following:
  • The radial distance $r$ typically ranges from a minimum value $r_1$ to a maximum value $r_2$
  • The polar angle $\theta$ ranges from a minimum value $\theta_1$ to a maximum value $\theta_2$, often 0 to $\pi$
  • The azimuthal angle $\phi$ ranges from a minimum value $\phi_1$ to a maximum value $\phi_2$, often 0 to $2\pi$
• If the region has symmetry, the limits of integration may simplify:
  • For rotational symmetry about the $z$-axis, the limits for $\phi$ may reduce to 0 to $2\pi$
  • For reflection symmetry about the $xy$-plane, the limits for $\theta$ may reduce to 0 to $\pi/2$

Integration Techniques

• When evaluating triple integrals in spherical coordinates, use the following techniques:
  1. Integrate with respect to $r$ first, treating $\theta$ and $\phi$ as constants
  2. Integrate with respect to $\theta$ next, treating $\phi$ as a constant
  3. Integrate with respect to $\phi$ last
• If the integrand and region of integration have symmetry, simplify the integral:
  • For rotational symmetry about the $z$-axis, the integral with respect to $\phi$ may simplify to a constant multiple of $2\pi$
  • For reflection symmetry about the $xy$-plane, the integral with respect to $\theta$ may simplify to twice the integral from 0 to $\pi/2$
• Use substitution or other integration techniques as needed for each variable
  • For integrals involving trigonometric functions of $\theta$, substitution with $u = \cos\theta$ or $u = \sin\theta$ may be helpful
• If the integrand is a vector-valued function $\vec{F}(r, \theta, \phi)$, integrate each component separately
• Remember to include the Jacobian determinant $r^2 \sin\theta$ in the integral
• Evaluate the limits of integration in the order $d\phi$, $d\theta$, $dr$ to obtain the final result

Real-World Applications

• Triple integrals in spherical coordinates have numerous applications in science and engineering, including:
  • Calculating the volume of objects with spherical symmetry (spheres, ellipsoids, and regions bounded by spherical surfaces)
  • Determining the mass and center of mass of objects with non-uniform density distributions
  • Computing moments of inertia for objects rotating about an axis
  • Evaluating electric and gravitational potentials and fields for charge and mass distributions with spherical symmetry
  • Modeling heat and mass transfer in spherical systems (heat conduction in a sphere, diffusion in a spherical catalyst pellet)
• In physics, spherical coordinates are used to describe systems with spherical symmetry:
  • Electrostatics: electric potential and field of a charged sphere or shell
  • Gravitation: gravitational potential and field of a spherical mass distribution
  • Quantum mechanics: solving the Schrödinger equation for the hydrogen atom
• In engineering, spherical coordinates are used to analyze and design systems with spherical components:
  • Aerospace engineering: modeling the aerodynamics of spherical objects (satellites, spacecraft, and missiles)
  • Mechanical engineering: calculating the stress and strain distributions in spherical pressure vessels and bearings
  • Chemical engineering: modeling mass transfer and reaction kinetics in spherical catalyst pellets and packed bed reactors

Common Pitfalls

• When working with triple integrals in spherical coordinates, be aware of these common pitfalls:
  • Forgetting to include the Jacobian determinant $r^2 \sin\theta$ in the integral
  • Incorrectly setting up the limits of integration for $r$, $\theta$, and $\phi$
    • Make sure the limits are in the correct order and correspond to the appropriate variable
    • Ensure the limits cover the entire region of integration
  • Misinterpreting the angles $\theta$ and $\phi$
    • Remember that $\theta$ is measured from the positive $z$-axis, not from the $xy$-plane
    • Ensure that $\phi$ is measured in the $xy$-plane from the positive $x$-axis
  • Incorrectly transforming the integrand from Cartesian to spherical coordinates
    • Double-check the transformation equations and simplify the expression if possible
  • Mishandling improper integrals that arise from the Jacobian determinant $r^2 \sin\theta$
    • If the integrand is undefined at $r = 0$ or $\theta = 0$, consider using limit evaluation or l'Hôpital's rule
  • Overlooking symmetry in the region of integration or integrand
    • Identify any rotational or reflection symmetry to simplify the integral limits or integrand
  • Improperly evaluating the limits of integration or the final result
    • Make sure to substitute the limits in the correct order ($d\phi$, $d\theta$, $dr$) and simplify the expression

Practice Problems

1. Evaluate the triple integral $\iiint_E (x^2 + y^2 + z^2) \, dV$ over the sphere $x^2 + y^2 + z^2 \leq 4$ using spherical coordinates.

2. Find the volume of the region bounded by the cone $z = \sqrt{x^2 + y^2}$ and the sphere $x^2 + y^2 + z^2 = 4$ using spherical coordinates.

3. Calculate the moment of inertia of a solid sphere of radius $R$ and uniform density $\rho$ about an axis through its center using spherical coordinates.

4. Evaluate the triple integral $\iiint_E \sin(\phi) \, dV$ over the region $E$ bounded by the spherical surfaces $r = 1$, $r = 2$, and the cone $\theta = \pi/4$ using spherical coordinates.

5. Determine the center of mass of a solid hemisphere of radius $R$ with density $\rho(r, \theta, \phi) = kr$, where $k$ is a constant, using spherical coordinates.

6. Find the electric potential at a point $(x, y, z)$ due to a charged sphere of radius $R$ and uniform charge density $\rho$ using spherical coordinates.

7. Evaluate the triple integral $\iiint_E \frac{1}{1 + r^2} \, dV$ over the region $E$ bounded by the spheres $r = 1$ and $r = 2$ using spherical coordinates.