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Triple Integral Tetrahedron with Vertices Calculator

Reviewed by Calculator Editorial Team

This calculator computes triple integrals over a tetrahedron defined by its four vertices in 3D space. It calculates the volume of the tetrahedron and evaluates integrals of functions over this region.

How to Use This Calculator

To calculate a triple integral over a tetrahedron:

  1. Enter the coordinates of the four vertices of the tetrahedron in the input fields.
  2. Enter the function you want to integrate in the function field.
  3. Click "Calculate" to compute the integral and volume.
  4. Review the results and chart visualization.

Note: The calculator uses the standard triple integral formula for a tetrahedron. The vertices must form a valid tetrahedron (non-coplanar points).

Formula Used

The volume of a tetrahedron with vertices A, B, C, and D is given by:

V = (1/6) |(B - A) · [(C - A) × (D - A)]|

The triple integral of a function f(x,y,z) over the tetrahedron is:

∫∫∫ f(x,y,z) dV = ∫∫∫ f(x,y,z) dx dy dz

The calculator uses numerical integration methods to approximate the integral when an analytical solution is not available.

Worked Example

Consider a tetrahedron with vertices at:

  • A = (0, 0, 0)
  • B = (1, 0, 0)
  • C = (0, 1, 0)
  • D = (0, 0, 1)

To compute the integral of f(x,y,z) = x² + y² + z² over this tetrahedron:

  1. Enter the vertex coordinates in the calculator.
  2. Enter the function "x^2 + y^2 + z^2".
  3. Click "Calculate".

The calculator will display the volume of the tetrahedron and the value of the integral.

Interpreting Results

The calculator provides two main results:

  1. Volume: The volume of the tetrahedron in cubic units.
  2. Integral Value: The value of the triple integral of the specified function over the tetrahedron.

The chart visualization shows the tetrahedron in 3D space with the vertices marked.

For complex functions, the calculator uses numerical integration which may introduce small approximation errors. The results are accurate to within the specified tolerance.

Frequently Asked Questions

What is a tetrahedron?
A tetrahedron is a three-dimensional shape with four triangular faces, six edges, and four vertices.
How do I define a tetrahedron for this calculator?
You need to provide the coordinates of the four vertices in 3D space. The vertices must not be coplanar.
What functions can I integrate with this calculator?
The calculator accepts any mathematical expression involving x, y, and z. Common functions include polynomials, exponentials, and trigonometric functions.
Is the result always exact or can it be approximate?
The calculator uses numerical integration methods, so results are approximate. The accuracy depends on the integration tolerance settings.
Can I visualize the tetrahedron?
Yes, the calculator includes a 3D visualization of the tetrahedron with the vertices marked.