Triple Integral Calculator

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Triple Integral Calculator
Cartesian (x, y, z)
Define Integration Limits
to
to
to
Enter a constant (e.g., 1 for volume or density value)
Solution:
Result =
function calculateTripleIntegral(){var xmin=parseFloat(document.getElementById('x_min').value);var xmax=parseFloat(document.getElementById('x_max').value);var ymin=parseFloat(document.getElementById('y_min').value);var ymax=parseFloat(document.getElementById('y_max').value);var zmin=parseFloat(document.getElementById('z_min').value);var zmax=parseFloat(document.getElementById('z_max').value);var fVal=parseFloat(document.getElementById('integrand').value);if(isNaN(xmin)||isNaN(xmax)||isNaN(ymin)||isNaN(ymax)||isNaN(zmin)||isNaN(zmax)||isNaN(fVal)){alert('Please enter valid numeric values for all limits and the integrand.');return;}var dx = xmax – xmin;var dy = ymax – ymin;var dz = zmax – zmin;var volume = dx * dy * dz;var totalResult = volume * fVal;document.getElementById('resultValue').innerHTML = totalResult.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 4});if(document.getElementById('steps').checked){var stepsHTML = "1. Inner Integral (dx): ∫[" + xmin + " to " + xmax + "] " + fVal + " dx = " + (fVal * dx).toFixed(4) + "
";stepsHTML += "2. Middle Integral (dy): ∫[" + ymin + " to " + ymax + "] " + (fVal * dx).toFixed(4) + " dy = " + (fVal * dx * dy).toFixed(4) + "
";stepsHTML += "3. Outer Integral (dz): ∫[" + zmin + " to " + zmax + "] " + (fVal * dx * dy).toFixed(4) + " dz = " + totalResult.toFixed(4);document.getElementById('stepDetails').innerHTML = stepsHTML;document.getElementById('stepDetails').style.display = 'block';} else {document.getElementById('stepDetails').style.display = 'none';}document.getElementById('calculatorAnswer').style.display='block';}

Triple Integral Calculator Use

The triple integral calculator is a specialized tool designed to solve multivariable calculus problems involving three-dimensional regions. While double integrals help find areas and volumes under surfaces, triple integrals are essential for calculating the properties of 3D solids, such as volume, total mass, and center of gravity.

This calculator currently supports rectangular (Cartesian) coordinate systems with constant bounds, which is the foundational method for evaluating integrals of the form ∫∫∫ f(x,y,z) dV.

x, y, and z Limits
These represent the boundaries of the solid region in the three-dimensional space. The "min" is the lower bound and "max" is the upper bound for each axis.
Integrand f(x, y, z)
The function inside the integral. If you are calculating volume, set this value to 1. If you are calculating mass, enter the density of the object (as a constant).

How It Works

When you evaluate a triple integral over a rectangular box B, you are essentially summing up small pieces of information across the width, depth, and height of the object. The standard formula used by the triple integral calculator for rectangular bounds is:

Result = ∫z1z2 ∫y1y2 ∫x1x2 f(x, y, z) dx dy dz

  • dx, dy, dz: Represent the differential elements of length along the x, y, and z axes.
  • dV: The differential volume element (dV = dx dy dz).
  • Constant Integrand: If f(x,y,z) = C, the result is simply C × (Length × Width × Height).

Calculation Example

Example: Find the mass of a rectangular block where the density is a constant 5 kg/m³, with dimensions ranging from x=0 to 2, y=0 to 3, and z=0 to 4.

Step-by-step solution:

  1. Identify limits: x[0,2], y[0,3], z[0,4]
  2. Set f(x,y,z) = 5
  3. Calculate inner integral (x): ∫02 5 dx = [5x] from 0 to 2 = 10
  4. Calculate middle integral (y): ∫03 10 dy = [10y] from 0 to 3 = 30
  5. Calculate outer integral (z): ∫04 30 dz = [30z] from 0 to 4 = 120
  6. Result = 120 units

Common Questions

What is the difference between a double and triple integral?

A double integral calculates the area under a curve or the volume under a 2D surface. A triple integral calculates values within a 3D volume, such as the total mass of a solid with varying density or the volume of a 4D "hyper-volume" (though geometrically we usually use it for 3D properties).

Can I use this for Spherical coordinates?

Currently, this triple integral calculator focuses on Cartesian coordinates. For spherical coordinates, you must include the Jacobian factor (ρ² sinφ) in your integrand before calculating the bounds of ρ, θ, and φ.

Why is my result negative?

If your lower limit is greater than your upper limit, the integral property ∫ab = -∫ba applies, which can result in a negative value. Always ensure your "min" value is smaller than your "max" value for physical volume calculations.

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