Applications of Triple Integrals

Let’s begin with the special case where  for all points in E. Then the triple integral does represent the volume of E:

EXAMPLE 4: Use a triple integral to find the volume of the tetrahedron T bounded by the planes 

The tetrahedron T and its projection D onto the xy-plane are shown in Figures below 

The lower boundary of T is the plane  and the upper boundary is the plane , that is,  .

Therefore we have

All the applications of double integrals can be immediately extended to triple integrals. For example, if the density function of a solid object that occupies the region E is , in units of mass per unit volume, at any given point , then its mass is

and its moments about the three coordinate planes are

The center of mass is located at the point , where

If the density is constant, the center of mass of the solid is called the centroid of E. The moments of inertia about the three coordinate axes are

The total electric charge on a solid object occupying a region E and having charge density  is

If we have three continuous random variables X,Y and Z their joint density function is a function of three variables such that the probability that (X,Y,Z) lies in E is

In particular,

The joint density function satisfies

EXAMPLE 5: Find the center of mass of a solid of constant density that is bounded by the parabolic cylinder  and the planes , and .

The solid E and its projection onto the xy-plane are shown in Figure below.

The lower and upper surfaces of E are the planes  and  so we describe E as a type 1 region:

Then, if the density is  the mass is

Because of the symmetry of E and about the xz-plane, we can immediately say that  and therefore  . The other moments are

Therefore the center of mass is