Triple Integrals

How do rocket engineers compute the center of mass of a fuel-loaded Falcon 9 to within 0.5%? With a triple integral over a complicated 3D region.

• SpaceX Falcon 9: centroid and moment of inertia of fuel-loaded stages, computed as triple integrals
• GE Aviation: stress, temperature, airflow integrals over jet-engine compressor stages, cylindrical coordinates throughout
• NASA gravitational modelling: integrate mass density over irregular asteroids to predict satellite trajectories
• ANSYS Fluent CFD: combustion chamber concentration totals computed as triple integrals over millions of tetrahedral cells

Предварительные знания

• Double integrals and Fubini's theorem
• Polar coordinates and the Jacobian
• Visualisation of 3D solids

• Double Integrals

Triple Integrals in Cartesian Coordinates

ANSYS Fluent partitions a 3D combustion chamber into 10 million tetrahedral cells, evaluates fuel concentration in each, and sums to get the total mass (a triple integral). The same construction works for the moment of inertia of a turbine blade, the charge inside a capacitor, or the kinetic energy stored in a rotating flywheel.

Centroid coordinates: x_bar = (1/m) * triple integral of x rho dV, and similarly for y_bar, z_bar. Moments of inertia: I_z = triple integral of (x^2 + y^2) rho dV. All standard quantities from rigid-body mechanics are triple integrals.

Sketch the solid (or at least its xy-projection) before writing limits. The innermost variable is bounded by two surfaces; the middle by two curves on the projection; the outermost by two numbers.

Cylindrical Coordinates

GE Aviation models a jet engine compressor as 17 stacked cylindrical sections. Stress, temperature, and airflow integrals over each section are written in cylindrical coordinates because the geometry is naturally rotationally symmetric. Cartesian would multiply the code complexity by an order of magnitude.

Use cylindrical coordinates when the solid has an axis of symmetry along z (cylinders, cones, paraboloids, donut-like shapes). The factor r in the volume element often plays nicely with integrands of the form f(x^2 + y^2, z).

Common error: forgetting the Jacobian r. Without it, you compute a different (incorrect) integral. The rule: every time you switch to cylindrical, multiply by r.

Spherical Coordinates

NASA computes the gravitational pull of an asteroid on a satellite by integrating mass density over the asteroid's volume in spherical coordinates. SpaceX's Starlink team integrates antenna gain over the celestial sphere to optimise satellite-to-ground link budgets. Anything with spherical symmetry (gravity, electric fields, radar coverage) wants spherical coordinates.

Notation varies. Many physics texts swap the roles of phi and theta. The math convention used here is: phi = polar (from z-axis), theta = azimuthal. Always check the convention in any text you consult.

If the integrand depends on x^2 + y^2 + z^2 = rho^2, or if the region is a sphere/cone/spherical shell, switch to spherical immediately. The integrand simplifies and the volume element absorbs the geometric factor.

Mass, Centroid, and Moments of Inertia

When SpaceX designs the Falcon 9 second stage, they need the precise centroid and moment of inertia of every fuel-loaded configuration. Off by 2% on the centroid and the rocket loses attitude control during burn. Every one of those numbers is a triple integral over a 3D region, evaluated with high-fidelity density data.

Pappus's theorem (1st centuries): the volume of a solid of revolution equals the area of the generating region times the distance the centroid travels. A triple integral collapses to a one-line calculation if the geometry is symmetric.

Before choosing coordinates: examine symmetry. Rotational about z -> cylindrical. Spherical -> spherical. None -> Cartesian. Coordinate choice cuts work in half (or doubles it).

Where triple integrals lead next

Triple integrals are the natural domain of the divergence theorem and of nearly every quantitative model in continuum mechanics, electromagnetism, and gravitational physics.

• Change of variables — General Jacobian theory unifies cylindrical, spherical, and ad-hoc coordinate maps
• Divergence theorem — Converts a triple integral of div F over a solid into a surface integral over its boundary
• Continuum mechanics — Stress, strain, deformation are integrated over 3D bodies via triple integrals
• Probability in R^3 — Joint density of three random variables is integrated to get probabilities

Итоги

• Triple integral = limit of 3D Riemann sums: triple integral of f dV = lim sum f(x*, y*, z*) Delta x Delta y Delta z
• Cartesian iterated form: integrate over z between two surfaces, then over y between two curves, then over x between two numbers
• Cylindrical: x = r cos theta, y = r sin theta, z = z; dV = r dr d theta dz. Use for axial symmetry.
• Spherical: x = rho sin phi cos theta, y = rho sin phi sin theta, z = rho cos phi; dV = rho^2 sin phi d rho d phi d theta. Use for radial symmetry.
• Applications: volume = triple integral of dV; mass = triple integral of rho dV; centroid = (1/m) * triple integral of x rho dV; moment of inertia I_z = triple integral of (x^2 + y^2) rho dV

Связанные уроки

• calc-21-double-integrals — Triple integrals extend the double-integral construction by one dimension
• calc-23-coordinate-changes — Cylindrical and spherical coordinates are the workhorse change-of-variables for 3D
• calc-26-divergence-theorem — The divergence theorem turns a triple integral over a solid into a surface integral over its boundary