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Calculus III (Multivariable) Unit 6: Triple integrals & change of variables

Triple integrals extend to 3D regions. Cylindrical and spherical coordinates simplify regions with circular symmetry, using a Jacobian scale factor.

Covers: Cylindrical & spherical coordinates, Jacobians

Study guide

Triple integrals ∭_E f dV extend double integrals to solid regions. With f = 1 they give volume; with a density they give mass. Set them up as iterated integrals with the innermost limits depending on the outer variables, usually integrating z first between a bottom surface and a top surface, then over the region's shadow in the xy-plane.

Cylindrical coordinates (r, θ, z) use polar coordinates in the xy-plane plus height: x = r cos θ, y = r sin θ, z = z, with dV = r dz dr dθ. They suit cylinders, cones and regions with circular symmetry around the z-axis.

Spherical coordinates (ρ, θ, φ) use distance ρ from the origin, the angle θ around the z-axis, and the angle φ down from the positive z-axis (0 ≤ φ ≤ π). The conversions are x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ, and dV = ρ² sin φ dρ dφ dθ. They suit spheres, cones from the origin, and integrands involving x² + y² + z².

A general change of variables x = g(u, v), y = h(u, v) requires the Jacobian, the determinant ∂(x, y)/∂(u, v), and replaces dA with |Jacobian| du dv. The Jacobian measures how the transformation stretches area. The r in polar and the ρ² sin φ in spherical coordinates are their Jacobians.

In 3D, M = ∭ ρ dV and the center of mass is (x̄, ȳ, z̄), with z̄ = (1/M)∭ zρ dV and similar formulas for x̄ and ȳ. In spherical coordinates, the sphere x² + y² + z² = a² is ρ = a, the cone z = √(x² + y²) is φ = π/4, and the plane z = a is ρ = a sec φ.

Choose u and v so the region becomes a rectangle or a disc. For the ellipse x²/a² + y²/b² ≤ 1, the substitution x = au, y = bv gives the unit disc with Jacobian ab. If u and v are given in terms of x and y, compute ∂(u, v)/∂(x, y) and take its reciprocal.

Key ideas

• Set up dV in the order that makes limits simplest.
• Cylindrical: (r, θ, z), good for cylinders and cones.
• Spherical: (ρ, φ, θ), good for spheres.
• A general change of variables multiplies by the absolute value of the Jacobian determinant.

Formulas and rules

Common mistakes

• Forgetting ρ² sin φ in spherical coordinates.
• Letting φ run past π.

Words to know

dV in cylindrical coordinates

r dz dr dθ

dV in spherical coordinates

ρ² sin φ dρ dφ dθ

Range of φ in spherical coordinates

0 ≤ φ ≤ π

z in spherical coordinates

z = ρ cos φ

Jacobian

The factor that scales area or volume in a change of variables.

Triple integral

∭_E f dV over a solid region.

Cylindrical coordinates

(r, θ, z) with x = r cos θ, y = r sin θ.

Spherical coordinates

(ρ, θ, φ): distance, angle around z, angle down from +z.

x and y in spherical coordinates

x = ρ sin φ cos θ, y = ρ sin φ sin θ

Change of variables (dA)

dA = |∂(x, y)/∂(u, v)| du dv

Volume as a triple integral

∭_E 1 dV

Sample practice questions

19 more questions like these, from level 1 to 5, are in the practice set.