Triple Integrals: Evaluation in Cartesian, Cylindrical & Spherical Coordinates and Volume of Solids

Explanation

Triple integrals extend multi-variable accumulation across three-dimensional Euclidean volumes. When physical geometries exhibit radial symmetry about an axis, cylindrical transformations simplify boundaries to constant cylinder limits r = R. When geometries exhibit point symmetry about the origin (e.g., gravitating bodies, celestial spheres, electrical field potentials), spherical polar coordinates decouple boundaries into completely independent rectangular product limits [0, R] × [0, π] × [0, 2π].

Key points

• When $f(x,y,z)=1$, the triple integral yields the total solid Volume: $\text{Volume}(V)={\iiint }_{V}dxdydz$.
• Cartesian Evaluation: ${\iiint }_{V}fdV={\int }_{x=a}^b{\int }_{y=g_1(x)}^{g_2(x)}{\int }_{z=h_1(x,y)}^{h_2(x,y)}f(x,y,z)dzdydx$.
• Cylindrical Coordinates: $x=r\cos \theta$, $y=r\sin \theta$, $z=z$ with $dV=rdzdrd\theta$. Ideal for solids of revolution, cylinders, and cones.
• Spherical Polar Coordinates: $x=\rho \sin \varphi \cos \theta$, $y=\rho \sin \varphi \sin \theta$, $z=\rho \cos \varphi$ with $dV={\rho }^2\sin \varphi d\rho d\varphi d\theta$ (where $\rho \ge 0$, $0\le \varphi \le \pi$ is colatitude, $0\le \theta \le 2\pi$ is azimuth).
• Physical Applications: $\text{Mass }M={\iiint }_V\rho (x,y,z)dV$; Center of gravity $\bar{z}=\frac{1}{M}{\iiint }_{V}z\rho dV$; Moment of inertia $I_z={\iiint }_V(x^2+y^2)\rho dV$.

Example

Code examples

// Derivation: Volume of a Sphere via Spherical Triple Integration
// Sphere Equation: x^2 + y^2 + z^2 ≤ a^2
// Coordinate transformation:
//   x = ρ sin φ cos θ,  y = ρ sin φ sin θ,  z = ρ cos φ
//   Volume element: dV = ρ^2 sin φ dρ dφ dθ
// Limits for the entire sphere:
//   Radial distance: ρ ranges from 0 to a
//   Colatitude angle: φ ranges from 0 to π
//   Azimuthal angle:  θ ranges from 0 to 2π
//
// Volume V = ∫_0^{2π} dθ ∫_0^π sin φ dφ ∫_0^a ρ^2 dρ
// Step 1: Radial integral:
//   ∫_0^a ρ^2 dρ = [ ρ^3 / 3 ]_0^a = a^3 / 3
// Step 2: Colatitude integral:
//   ∫_0^π sin φ dφ = [ -cos φ ]_0^π = -(-1) - (-1) = 1 + 1 = 2
// Step 3: Azimuthal integral:
//   ∫_0^{2π} dθ = [ θ ]_0^{2π} = 2π
//
// Total Volume:
//   V = (2π) * (2) * (a^3 / 3) = (4/3) π a^3 (Q.E.D.)

Exam questions

• [8 Marks] Evaluate ∭_V dx dy dz / (x + y + z + 1)³ over the region bounded by x = 0, y = 0, z = 0, and x + y + z = 1.
• [8 Marks] Find the volume common to the cylinders x² + y² = a² and x² + z² = a² using multiple integration.
• [7 Marks] Find the center of gravity of a uniform hemisphere of radius a using spherical polar coordinates.