Triple Integral Calculator

Evaluate triple integral of xyz over [0,1] x [0,2] x [0,3]: Step 1: Inner (x): integral from 0 to 1 of xyz dx = (x^2/2)yz from 0 to 1 = yz/2 Step 2: Middle (y): integral from 0 to 2 of yz/2 dy = (y^2/4)z from 0 to 2 = z Step 3: Outer (z): integral from 0 to 3 of z dz = z^2/2 from 0 to 3 = 4.5 Result: 4.5

A triple integral is a powerful mathematical tool that extends the concept of single and double integrals into three dimensions. While a single integral calculates the area under a curve and a double integral finds the volume under a surface, a triple integral integrates a function f(x,y,z) over a three-dimensional region in space, often denoted as dV or dx dy dz. This allows us to compute various physical quantities associated with 3D objects. For instance, if f(x,y,z) represents the density of an object at a given point, the triple integral of f over the object's volume will yield its total mass. If f(x,y,z) is simply 1, the triple integral calculates the volume of the region itself. The evaluation process for a triple integral over a rectangular box involves iterated integration, meaning you integrate with respect to one variable at a time, treating the others as constants, and then evaluate the result at its bounds, repeating for all three variables in sequence. This method is particularly straightforward for polynomial functions, where the power rule of integration is applied repeatedly.