Calculadora Integral Tripla

Triple integrals are used to calculate volumes, masses, and other physical quantities in three-dimensional space. This calculator helps you compute triple integrals of functions with respect to x, y, and z.

What is a Triple Integral?

A triple integral extends the concept of double integrals to three dimensions. It's used to calculate quantities like volume, mass, and charge density in three-dimensional regions. The general form is:

∫∫∫ f(x,y,z) dV = ∫[c][d] ∫[a][b] ∫[α][β] f(x,y,z) dx dy dz

The limits of integration define the region over which the function is integrated. The order of integration can vary depending on the problem.

How to Use This Calculator

Enter the function you want to integrate in the function field (e.g., x² + y² + z²)
Specify the limits of integration for x, y, and z
Select the order of integration (x-y-z, x-z-y, etc.)
Click "Calculate" to compute the integral
View the result and visualization

Formula

The triple integral of a function f(x,y,z) over a region D is calculated as:

∫∫∫ f(x,y,z) dV = ∫[z1][z2] ∫[y1][y2] ∫[x1][x2] f(x,y,z) dx dy dz

Where:

f(x,y,z) is the integrand function
x1 and x2 are the lower and upper limits for x
y1 and y2 are the lower and upper limits for y
z1 and z2 are the lower and upper limits for z

Worked Example

Let's calculate the volume under the plane z = 2 - x - y from x=0 to x=1, y=0 to y=1, and z=0 to z=2-x-y.

∫[0][1] ∫[0][1] ∫[0][2-x-y] 1 dz dy dx

The result is 1/3 cubic units.

Applications

Triple integrals are used in various fields including:

Physics for calculating mass and charge distributions
Engineering for volume calculations
Probability for multivariate distributions
Fluid dynamics for flow calculations

FAQ

What is the difference between single, double, and triple integrals?: Single integrals calculate areas under curves, double integrals calculate volumes under surfaces, and triple integrals calculate volumes in three-dimensional space.
When would I use a triple integral instead of a double integral?: Use triple integrals when working with three-dimensional regions or quantities that depend on three variables.
What are the common limits of integration for triple integrals?: Common limits include rectangular prisms, spheres, and other simple three-dimensional shapes.
Can this calculator handle improper integrals?: This calculator is designed for proper triple integrals with finite limits. Improper integrals would require special handling.
What if my function is not continuous?: The calculator assumes the function is continuous over the integration region. Discontinuous functions may require special treatment.