Triple Integrals

Introduction

Integrals represent area under a curve. Double integrals represent volume under a surface. Triple integrals represent a 4 dimensional hyper-volume under a 3 dimensional volume.

The Essentials

A triple integral looks like this:

\[ \iiint_E f(x,y,z) dV \]

For rectangular coordinates \( dV=dxdydz \). If the region to be integrated over isn't a rectangular box, the bounds can be functions of the other variables. The inside bound can be a function of the variables of the middle and outside variables, and the bound of the middle integral can be a function of the variable on the outside bound.

For example, if you wanted to integrate a function over the bound of the shape made with the vertices (0, 0, 0), (0, 1, 0), (1, 0, 0), (0, 0, 1), you could set it up like this:

\[ \int_0^1\int_0^{1-x}\int_0^{1-x-y}f(x,y,z)dzdydx \]

Where the first bound of integration goes from the xy-plane (\( z=0 \)) to the plane \( z=1-x-y \). After that the shape can be projecting onto the xy-plane and the bounds of the remaining two integrals can describe the 2D shape. In this case, it is a triangle with the vertices (0, 0), (0, 1), (1, 0). The middle integral, therefore, has the bounds from \( y=0 \) to \( y=1-x \) and the last integral from \( x=0 \) to \( x=1 \).

When integrating over cylindrical coordinates \( dV=rdrd\theta dz \). When integrating over spherical coordinates \( dV=\rho^2\sin(\phi)d\rho d\theta d\phi \)

Example

Evaluate the triple integral of the function \( f(x,y,z)=\sqrt{x^2+y^2+z^2} \) over the region defined by the first octant of a sphere centered at the origin with a radius of 3.

Converting the function to integrate to spherical coordinates gives us:

\[ f(x,y,z)dV=\rho^3\sin(\phi)d \rho d\theta d\phi \]

To get the octant of the sphere, \( \rho \) will go from 0 to 3, \( \theta \) will go from 0 to \( \frac{\pi}{2} \), and \( \phi \) will go from 0 to \( \frac{\pi}{2} \):

\[ \int_0^{\frac{\pi}{2}}\int_0^{\frac{\pi}{2}}\int_0^3 \rho (\rho^2\sin(\phi))d\rho d\theta d\phi \]

Because the functions are separable and the bounds are constant, we can separate the triple integral into three single integrals:

\[ \int_0^{\frac{\pi}{2}}\sin(\phi)d\phi \int_0^{\frac{\pi}{2}}d\theta \int_0^3 \rho^3d\rho \]
\[ \frac{81\pi}{8} \]

Practice

Evaluate the triple integral of the function \( f(x,y,z)=x^2+xy+xz \) with the bounds being the points (0, 0, 0), (2, 2, 0), (2, 2, 1),(2, 0, 1), and (2,0,0).

Solution:

5.6

More Resources

Openstax: Triple integrals External Link Icon

Openstax: Triple integrals in Cylindrical and Spherical Coordinates External Link Icon