Triple Integral
A triple integral extends the concept of integration to three dimensions, computing the accumulation of a function over a three-dimensional region. It is used to find volumes, masses, and other physical quantities distributed throughout a solid region in space. Applications include calculating mass of a solid with variable density, electric charge distributions, and gravitational potential fields.
Key Formula
∫∫∫_V f(x,y,z) dx dy dz
LaTeX: \iiint_V f(x,y,z)\,dx\,dy\,dz
| Symbol | Meaning | Unit |
|---|---|---|
| f(x,y,z) | Integrand function in three variables | varies |
| V | Three-dimensional region of integration | dimensionless (region) |
| dx dy dz | Volume element in Cartesian coordinates | m³ or appropriate unit |
Worked Example
Problem
Find the volume of the rectangular box defined by 0 ≤ x ≤ 2, 0 ≤ y ≤ 3, 0 ≤ z ≤ 4 using a triple integral.
Solution
Step 1: Set up the triple integral for volume: V = ∫₀² ∫₀³ ∫₀⁴ 1 dz dy dx. Step 2: Integrate with respect to z: ∫₀⁴ 1 dz = [z]₀⁴ = 4. Step 3: Integrate with respect to y: ∫₀³ 4 dy = [4y]₀³ = 12. Step 4: Integrate with respect to x: ∫₀² 12 dx = [12x]₀² = 24.
Answer
Volume = 24 cubic units
Coordinate Systems for Triple Integrals
| Coordinate System | Volume Element | Best Used For | Example Region |
|---|---|---|---|
| Cartesian | dx dy dz | Rectangular boxes, cuboids | Box: a≤x≤b, c≤y≤d, e≤z≤f |
| Cylindrical | r dr dθ dz | Cylinders, cones, paraboloids | Cylinder: r≤R, 0≤z≤h |
| Spherical | ρ² sin φ dρ dφ dθ | Spheres, hemispheres | Sphere: ρ≤R |
| General | det(J) du dv dw | Irregular regions via substitution | Any region with Jacobian |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
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.
Surface Integral
A surface integral extends integration to a two-dimensional surface embedded in three-dimensional space, computing the total value of a scalar or vector field over that surface. Scalar surface integrals find quantities like surface area or total mass of a thin shell, while vector surface integrals (flux integrals) measure how much of a vector field passes through a surface. Surface integrals are central to Maxwell's equations, fluid dynamics, and the theorems of Gauss and Stokes.
From Latin "integrare" meaning "to make whole" or "to complete". The concept of integration was developed independently by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. The triple integral is a natural extension, formalized in the 18th–19th centuries.