Triple Integrals - Integral Calculus of Multivariable Functions - University Mathematics Handbook

University Mathematics Handbook (2015)

VIII. Integral Calculus of Multivariable Functions

Chapter 6. Triple Integrals

6.1  Definition

Let function be defined above solid in three-dimensional space of volume . Let us divide into elementary solids , of volumes , respectively. Let's choose, in every , point and construct integral sum

If there exists , when the maximum diameter of solids tends to zero and the limit is independent of the partition of and the choice of points , then it is called triple integral of over and is denoted as either

or or

6.2  Properties of Triple Integrals

The properties of triple integrals are similar to those of double integrals (see chap.2.2). Let us just mention the following:

a.  Volume of solid is .

b.  If function is integrable over , and holds for every , then

c.  If function is integrable over , then is integrable in the same domain, and

6.3  Triple Integrals Calculation

a.  If function is integrable over rectangular parallelepiped ,

then

b.  If function is integrable above cylindrical body bounded between two surfaces , above region on plane (Figure 1), then

Figure 1

Figure 2

c.  If body is between planes and , and for every constant of the plane parallel to these planes carves area from body (Figure 2), then

.

6.4  Change of Variables in Triple Integrals

a.  If the system of functions , , , and , continuous and with continuous partial derivatives, maps to body with one-to-one correspondence and the Jacobian

is different from zero, then

b.  In cylindrical coordinates , , , the Jacobian is

c.  In spherical coordinates: ,

the Jacobian is

6.5  Applications of Triple Integrals

a.  Volume of the body: .

b.  Mass of a body consisting of a material with specific gravity is .

c.  Static moments of inertia , , in relation to planes , , and , respectively, is

, ,

d.  Center of mass of body :

when are static moment calculated using the formulas mentioned in paragraph c, and m is the mass of the body calculated by formula mentioned in paragraph b.