Triple Integrals in Spherical Coordinates

The central idea

Using the convention $\rho\ge0$, azimuth $0\le\theta<2\pi$, and polar angle $0\le\phi\le\pi$ measured from $+z$, $x=\rho\sin\phi\cos\theta$, $y=\rho\sin\phi\sin\theta$, $z=\rho\cos\phi$, and $dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$. Conventions for $\theta,\phi$ must be stated.

Definitions, hypotheses, and notation

A cone $z=\sqrt{x^2+y^2}$ becomes $\rho\cos\phi=\rho\sin\phi$, hence $\phi=\pi/4$ on standard ranges. A sphere not centered at the origin may produce a radial bound such as $\rho=2a\cos\phi$ after completing the geometry. Coordinate choice helps most when boundaries align with coordinate surfaces.

The origin and polar axis are coordinate singularities where angles are nonunique, but they form measure-zero sets. Correct coverage and the Jacobian keep the integral valid. For density depending only on distance from the origin, angular integrations often separate completely.

Conceptual meaning

A spherical cell expands in two angular directions: one factor $\rho$ from polar arc length and another $\rho\sin\phi$ from azimuthal arc length. Multiplying by radial thickness gives the Jacobian $\rho^2\sin\phi$.

A dependable method and decision rule

1. State the angle convention and sketch radial, polar, and azimuthal restrictions.
2. Write $\rho$ bounds from origin to bounding surfaces.
3. Convert $x^2+y^2+z^2$ to $\rho^2$ and $z$ to $\rho\cos\phi$.
4. Include $\rho^2\sin\phi$.
5. Check that angle ranges cover the solid once and respect hemispheres or cones.

Fully worked example

Graphical or geometric meaning

Constant $\rho$ gives spheres, constant $\phi$ gives cones about the $z$-axis, and constant $\theta$ gives vertical half-planes. A hemisphere changes the $\phi$ range, not necessarily $\rho$.

Common mistakes and why they fail

Verification and reasonableness checks

• Integrate one over a ball and recover $4\pi a^3/3$.
• Test coordinate formulas at the north pole and equator.
• Verify Jacobian is nonnegative on the chosen standard ranges.

Fix the angle convention before setting bounds

Using the common convention, $\rho$ is distance from the origin, $\phi$ is measured from the positive $z$-axis, and $\theta$ is the azimuth in the $xy$-plane. Then $dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$. Some texts swap angle names, so state the convention before interpreting bounds. Spheres give constant $\rho$, cones give constant $\phi$, and vertical half-planes give constant $\theta$. Convert $x^2+y^2+z^2$ to $\rho^2$ and retain both Jacobian factors. Because $\sin\phi\ge0$ on $0\le\phi\le\pi$, standard bounds preserve positive volume. Test the full-sphere case with integrand one to confirm angular coverage and normalization.

Practice

1. What $\phi$ range gives the upper hemisphere?
2. Convert $x^2+y^2+z^2=9$.
3. What is the spherical Jacobian?

Connections and next steps

• Triple Integrals
• Triple Integrals in Cylindrical Coordinates
• Three-Dimensional Coordinate System