Double Integral
A double integral is the extension of a definite integral to functions of two variables, computing the cumulative sum of a function f(x, y) over a two-dimensional region D in the xy-plane. Geometrically, the double integral of a non-negative function gives the volume under the surface z = f(x, y) above the region D. Double integrals are evaluated using iterated integration (Fubini's theorem), and are applied to compute areas, volumes, masses of planar objects, centre of mass, moments of inertia, and probabilities in joint distributions.
Key Formula
∬_D f(x,y) dA = ∫_a^b ∫_(g₁(x))^(g₂(x)) f(x,y) dy dx
LaTeX: \iint_D f(x,y)\,dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx
| Symbol | Meaning | Unit |
|---|---|---|
| ∬_D | Double integral over region D | depends on f |
| f(x, y) | Integrand function of two variables | depends on context |
| dA | Area element (= dx dy in Cartesian coordinates) | unit area |
| a, b | Outer limits of integration (in x) | dimensionless |
| g₁(x), g₂(x) | Inner limits of integration (functions of x) | dimensionless |
Worked Example
Problem
Evaluate ∬_D (x + 2y) dA over the rectangle D = {0 ≤ x ≤ 2, 0 ≤ y ≤ 1}.
Solution
Step 1: Set up iterated integral: ∫₀² ∫₀¹ (x + 2y) dy dx. Step 2: Inner integral w.r.t. y (x fixed): ∫₀¹ (x + 2y) dy = [xy + y²]₀¹ = (x + 1) − 0 = x + 1. Step 3: Outer integral w.r.t. x: ∫₀² (x + 1) dx = [x²/2 + x]₀² = (2 + 2) − 0 = 4.
Answer
∬_D (x + 2y) dA = 4
Comparison of Double Integral Coordinate Systems
| System | Area Element dA | Limits | Best For |
|---|---|---|---|
| Cartesian (x, y) | dx dy | Rectangular bounds | Rectangles, simple regions |
| Polar (r, θ) | r dr dθ | 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π | Circles, symmetric regions |
| Type I region | dy dx | a≤x≤b, g₁(x)≤y≤g₂(x) | Regions bounded by functions of x |
| Type II region | dx dy | c≤y≤d, h₁(y)≤x≤h₂(y) | Regions bounded by functions of y |
| Fubini switch | Swap order | Re-derive bounds | Simplify hard inner integrals |
Interactive Tools
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Related Terms
Improper Integral
An improper integral is a definite integral that has either an infinite limit of integration or an integrand that becomes unbounded (has a vertical asymptote) within the interval of integration. These integrals are evaluated using limits, replacing the problematic bound with a parameter and taking the limit as the parameter approaches the infinity or the singularity. Improper integrals arise frequently in probability theory, Fourier analysis, and physics when computing total accumulated quantities over unbounded domains.
Partial Derivative
A partial derivative is the derivative of a multivariable function with respect to one variable while all other variables are held constant, measuring the function's instantaneous rate of change in a single coordinate direction. Denoted by the symbol ∂ (a stylized "d"), partial derivatives generalize single-variable differentiation to functions of two or more variables. They are essential in multivariable calculus, optimization, thermodynamics, fluid dynamics, and the formulation of partial differential equations.
Gradient Vector
The gradient of a scalar function is a vector field whose components are the partial derivatives of the function with respect to each independent variable, pointing in the direction of the steepest rate of increase of the function. Denoted ∇f (nabla f), the gradient generalizes the ordinary derivative to multivariable functions. The gradient is fundamental in optimization (gradient descent), physics (conservative force fields), and machine learning for training neural networks.
From Latin "duplus" (twofold) + "integralis" (making whole). The generalization of integration to multiple variables was developed by mathematicians including Leonhard Euler and Joseph-Louis Lagrange in the 18th century. Guido Fubini formalized the theorem linking double integrals to iterated single integrals in 1907.