Double Integrals: Explained with Examples, Notes and FAQs

A single integral adds up the area under a curve. But what if the quantity you care about spreads over a flat region — like the total mass of a metal plate whose density varies from point to point, or the volume trapped under a curved surface? Slicing in one direction is not enough; you need to accumulate over two dimensions at once.

A double integral, written &iint;R f(x, y) dA, adds up the values of f over every tiny patch dA of a region R. When f(x, y) ≥ 0, it equals the volume under the surface z = f(x, y) above R; when f = 1, it equals the area of R. In practice you compute it as an iterated integral: integrate with respect to one variable first (treating the other as constant), then integrate the result with respect to the second variable.

Key Concepts

• dA = dy dx (or dx dy) — a tiny rectangle of area.
• Iterated integral: ∫ab ∫cd f(x, y) dy dx means integrate in y first, then x, working inside out.
• Fubini’s theorem: for continuous f, the order of integration can be swapped — but the limits must be rewritten to describe the same region.
• Type I region: between curves y = g₁(x) and y = g₂(x); Type II: between x = h₁(y) and x = h₂(y).
• Properties: linearity, additivity over split regions, and &iint;R dA = area of R.

Formulas

• &iint;R f dA = ∫ab ∫g₁(x)g₂(x) f(x, y) dy dx
• Volume under surface: V = &iint;R f(x, y) dA (for f ≥ 0).
• Average value of f over R: (1 / area(R)) · &iint;R f dA.
• Polar form: &iint;R f dA = &iint; f(r cos θ, r sin θ) r dr dθ.

Worked Example

Evaluate ∫02 ∫13 (x + 2y) dy dx.

1. Inner integral (y from 1 to 3, x held constant): ∫13 (x + 2y) dy = [xy + y²]13.
2. Evaluate: at y = 3 we get 3x + 9; at y = 1 we get x + 1. Subtract: (3x + 9) − (x + 1) = 2x + 8.
3. Outer integral (x from 0 to 2): ∫02 (2x + 8) dx = [x² + 8x]02.
4. Evaluate: at x = 2 we get 4 + 16 = 20; at x = 0 we get 0. Answer: 20.
5. Cross-check by swapping the order: ∫13 ∫02 (x + 2y) dx dy = ∫13 (2 + 4y) dy = [2y + 2y²]13 = (6 + 18) − (2 + 2) = 20. It agrees.

Common Misconceptions

• “The order of integration never matters.” For continuous f on a rectangle the value is the same, but the limits must be rewritten correctly when you swap — blindly swapping limits on a non-rectangular region gives wrong answers.
• “dA is just dx dy multiplied.” In polar coordinates dA = r dr dθ; the Jacobian factor r is essential.
• “A double integral is always a volume.” It is a signed volume — parts of f below zero subtract from the total.

Real-World Applications

• Mass of a plate: mass = &iint; (density) dA when density varies across the plate.
• Probability: total probability over a 2D region from a joint density function.
• Engineering: fluid flow, electric charge, or heat distributed over a flat region.
• Mechanics: center of mass and moments of inertia of flat objects.

A double integral is just two single integrals nested inside each other: integrate in one variable, then integrate the result in the other. Set the limits to trace out the region exactly, keep the Jacobian factor (like r in polar) whenever you change coordinates, and remember the result is a signed volume — positive above the plane, negative below it.