24
24
cubic units
2
3
4
24
mass units
24
24
cubic units
2
3
4
24
mass units
The Triple Integral Calculator computes the exact triple integral of a constant function $$f(x,y,z) = k$$ over a rectangular box (parallelepiped) $$[x_1, x_2] \times [y_1, y_2] \times [z_1, z_2]$$. Triple integrals extend integration to three dimensions, computing hypervolumes, masses, and other accumulated quantities over solid regions in three-dimensional space.
The triple integral $$\iiint_B f(x,y,z)\,dV$$ generalizes the double integral by adding a third dimension. Just as double integrals sum values over a planar region, triple integrals sum values over a three-dimensional solid. The notation $$dV = dx\,dy\,dz$$ represents an infinitesimal volume element.
For a constant integrand $$f(x,y,z) = k$$ over a rectangular box, the triple integral simplifies elegantly. By Fubini's Theorem for three variables, the iterated integration yields $$k \cdot \Delta x \cdot \Delta y \cdot \Delta z$$, which is simply the constant multiplied by the volume of the box. While this may seem straightforward, it forms the foundation for understanding more complex triple integrals.
The physical interpretation depends on what $$k$$ represents. If $$k$$ is a mass density (mass per unit volume), the triple integral gives the total mass of the solid. If $$k = 1$$, the integral gives the geometric volume of the region. In electrostatics, if $$k$$ is a charge density, the integral gives the total charge. In fluid mechanics, if $$k$$ is a fluid density, the integral gives the total mass of fluid in the region.
This calculator provides the integral value, the box volume, the dimensions along each axis, and a mass interpretation (treating the constant as a uniform density). It serves as a pedagogical tool for understanding how triple integrals work over the simplest possible region and integrand, building intuition before tackling variable-density or non-rectangular problems.
Triple integrals are essential in engineering for computing moments of inertia of solid objects, in physics for gravitational and electromagnetic field calculations, in thermodynamics for total energy in a volume, and in probability for joint distributions of three random variables. Enter the constant value and the box bounds to compute the exact result.
The calculator applies Fubini's Theorem for triple integrals over a rectangular box:
$$\iiint_B k\,dV = \int_{x_1}^{x_2} \int_{y_1}^{y_2} \int_{z_1}^{z_2} k\,dz\,dy\,dx$$
Inner integral (z):
$$\int_{z_1}^{z_2} k\,dz = k(z_2 - z_1) = k \cdot \Delta z$$
Middle integral (y):
$$\int_{y_1}^{y_2} k \cdot \Delta z\,dy = k \cdot \Delta z \cdot (y_2 - y_1) = k \cdot \Delta z \cdot \Delta y$$
Outer integral (x):
$$\int_{x_1}^{x_2} k \cdot \Delta z \cdot \Delta y\,dx = k \cdot \Delta z \cdot \Delta y \cdot (x_2 - x_1) = k \cdot \Delta x \cdot \Delta y \cdot \Delta z$$
The final result is simply the constant times the volume of the box: $$\iiint_B k\,dV = k \cdot V_{\text{box}}$$.
The Triple Integral Value equals the constant multiplied by the box volume. When the constant is 1, this reduces to the geometric volume of the box. When the constant represents a physical density, the result is the total mass (or charge, energy, etc.) contained in the region.
The Box Volume is the product of the three side lengths $$\Delta x \cdot \Delta y \cdot \Delta z$$. This is the measure of the region of integration.
The Mass Interpretation treats the constant as a uniform density and gives the total mass. For uniform density, mass equals density times volume, which is exactly what the triple integral computes.
Inputs
Results
With f = 1, the triple integral simply gives the volume: 2 × 3 × 4 = 24 cubic units. This is the most basic application of a triple integral.
Inputs
Results
Box volume = 3 × 2 × 5 = 30 m³. Mass = 2.5 × 30 = 75 kg. The uniform density makes this a straightforward multiplication.
A triple integral $$\iiint_B f(x,y,z)\,dV$$ sums the values of a function $$f$$ over a three-dimensional region $$B$$. It generalizes single integrals (over intervals) and double integrals (over planar regions) to solid regions in 3D space.
A constant function represents uniform density, the simplest and most common case in many applications. It provides exact closed-form results and builds intuition for triple integration. Variable integrands (e.g., $$f = xyz$$) require more complex formulas but follow the same Fubini's Theorem approach.
Fubini's Theorem states that if $$f$$ is continuous on a rectangular box, the triple integral can be evaluated as three nested single integrals in any order: $$\iiint_B f\,dV = \int \int \int f\,dz\,dy\,dx$$. The order of integration (which variable first) does not affect the result.
Yes, but the limits of integration become functions of the other variables. For example, integrating over a sphere requires expressing limits in spherical coordinates or using variable bounds. This calculator handles rectangular boxes for exact closed-form computation.
Triple integrals compute total mass from density functions, moments of inertia for rotating bodies, gravitational potential energy, total electric charge in a region, average temperature in a solid, and probability for three-dimensional joint distributions.
Besides Cartesian coordinates $$(x, y, z)$$, triple integrals can use cylindrical coordinates $$(r, \theta, z)$$ or spherical coordinates $$(\rho, \theta, \phi)$$. The choice depends on the region's geometry. This calculator uses Cartesian coordinates, which are natural for rectangular boxes.
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