Double & Triple Integrals

Multiple integrals add up quantities over areas and volumes. Learn double and triple integrals, iterated integration and Fubini's theorem, setting limits for general regions, changing the order of integration, polar/cylindrical/spherical coordinates, and applications to area, volume, mass, centroids and moments of inertia.

Mass of a plate with varying thickness, heat stored in a block, second moment of area of a beam section, total charge in a region — all require integrating over 2D or 3D domains.

An analogy

A double integral is like counting all the sand on a beach: multiply sand depth by each small patch of area and add up every patch.

Iterated integrals

Integrate the inner variable first, treating the outer as constant, then integrate the result. For rectangles the order doesn't matter (continuous f).

General regions

Sketch the region; for type I, x runs between constants and y between curves. Changing order can turn an impossible integral into an easy one (e.g. ∫∫ e^(y²)).

Physical applications

Area (f = 1), volume under a surface, mass with variable density, centroids, moments of inertia for rotating parts, and second moments of area for beam bending.

Curvilinear coordinates

Use polar, cylindrical or spherical coordinates with the Jacobian for round domains. Numerical cubature (Gauss points) handles irregular shapes — the basis of FEM integration.

Formulas

• Iterated (Fubini): ∬_R f dA = ∫ₐᵇ ∫_{g₁(x)}^{g₂(x)} f(x, y) dy dx
• Mass & centroid: m = ∭ ρ dV, x̄ = (1/m) ∭ xρ dV
• Second moment (rectangle b × h): I = ∬ y² dA = bh³/12
• Polar: ∬ f dA = ∫∫ f(r cosθ, r sinθ) r dr dθ

Worked example: Rectangle

Given: ∫₀¹ ∫₀² (x + y) dy dx

Find: Value.

• Inner: [xy + y²/2]₀² = 2x + 2
• Outer: ∫₀¹(2x + 2) dx = 1 + 2

Answer: 3

Worked example: Second moment of area

Given: Rectangle b = 0.1 m, h = 0.2 m about centroid

Find: I.

• I = ∫_{−h/2}^{h/2} y² b dy = bh³/12
• = 0.1 × 0.008/12

Answer: 6.67 × 10⁻⁵ m⁴

Common mistakes

• Wrong: Constant limits for curved regions. — Right: Inner limits may depend on the outer variable.
• Wrong: Forgetting the Jacobian in polar. — Right: dA = r dr dθ.

Applications

• Beam design: Second moments of area.
• Rotating machinery: Moments of inertia.
• Heat & mass: Total energy or mass in a region.

Key takeaways

• Integrate inner variable first.
• Limits follow the region's shape.
• Mass, centroid, inertia via ∬/∭.
• Use polar/cylindrical/spherical + Jacobian.

Going further

• Gaussian cubature: Numerical integration in FEM.
• Monte Carlo integration: High-dimensional integrals.
• Parallel-axis theorem: I = I_c + Ad².