What Is a Triple Integral?
In multivariable calculus, a triple integral is an extension of a definite integral to functions of three variables, `f(x, y, z)`. While a single integral calculates the area under a curve and a double integral calculates the volume under a surface, a triple integral can be thought of as calculating the "hypervolume" under a four-dimensional object. More practically, it's used to sum up a quantity over a three-dimensional region. If the function `f(x, y, z)` represents the density of an object, the triple integral over that object's region gives its total mass. If the function is simply `1`, the triple integral gives the volume of the region. This triple integral calculator provides a numerical solution for definite triple integrals over a rectangular box.
Evaluating a Triple Integral
A triple integral is solved as an "iterated integral," meaning you integrate with respect to one variable at a time, from the inside out. The general form is:
The process involves three successive integrations:
- Innermost Integral: First, you integrate the function `f(x, y, z)` with respect to the innermost variable (e.g., `x`), treating `y` and `z` as constants.
- Middle Integral: Next, you integrate the result of the first step with respect to the middle variable (e.g., `y`), treating `z` as a constant.
- Outermost Integral: Finally, you integrate the result of the second step with respect to the last variable (e.g., `z`). The final result is a single number.
While symbolic integration can be complex, this calculator uses a numerical method called Simpson's rule to approximate the value of the definite integral.
Applications of Triple Integrals
Triple integrals are a fundamental tool in physics, engineering, and other sciences for describing properties of three-dimensional objects.
- Calculating Volume: By setting `f(x, y, z) = 1`, the triple integral simply calculates the volume of the region defined by the integration bounds. For regular shapes, you can confirm this with our Volume Calculator.
- Finding Mass and Center of Mass: If you have a function `ρ(x, y, z)` that describes the density of an object at any point, the triple integral of the density function over the object's volume will give its total mass.
- Electromagnetism: In physics, triple integrals are used to calculate quantities like total electric charge or magnetic flux within a three-dimensional space.
For a deeper dive into the theory and techniques of multivariable calculus, educational resources like MIT OpenCourseWare's Multivariable Calculus course offer comprehensive lectures and materials.