The triple integral extends the concept of a double integral by summing up rectangular solids of sides Dx, Dy, and Dz. The volume of a solid defined by two surfaces, g1( x, y ) and g2( x, y ), over a region in the xy plane, f1( x ) £ y £ f2( x ) and a £ x £ b, is:.
Please note that the order of integration could be changed to any of the six possible permutations of dz, dy and dx. For example to find the volume of the solid bounded by the surfaces z = x2 + y2 and z = 8 - x2 - y2,
we will integrate from bottom surface to top, find the
intersection of the two surfaces, project it down to the xy plane and
integrate over that region. The resulting triple integral is: 
.
The intersection is the circle of radius 2 centered at the origin.
Use the symmetry of the situation to integrate over the first
quadrant and multiply by 4. Of course the volume could be done with
polar coordinates which are called cylindrical coordinates in
3-space. They are the same polar coordinates from before with a z
coordinate, ( r, ø, z ). Using cylindrical coordinates the
volume can be evaluated with: 
.
This triple integral is simple enough for us to do by hand. Verify
that the volume is 16p.
If the density of this solid were equal to the z coordinate, the
mass of the solid could be calculated by either integral: 
In other words, to find the mass integrate the density function just like we did before.
To find the center of mass, find the first moment from each coordinate plane and divide by the mass.
In this example, to find the z coordinate of the center of mass,
divide the moment from the xy plane which is the height above the xy
plane, z , times the density, z , by the mass:
Of course we could have done the problem in cylindrical coordinates. A good exercise is to do just that and compare the answers.
Spherical coordinates
Spherical coordinates are another way to represent a point in 3-space, where r is the distance to the origin, q is the usual angle from the x-axis, and f is the angle from the positive z-axis.
The formulas are used to change from rectangular to spherical and vice versa. The integration factor for spherical coordinates is r2sin( f ).
To find the volume of the solid bounded by the top half of the sphere x2 + y2 + z2 = 9 and the cone z = ÷( 3x2 + 3y2 ):
