Triple Integrals

    First, triple integrals are defined for boxed regions and their generalization from the Riemann integral of a one and two variable function is described in detail. The relationship between triple integrals and iterated integrals (Fubini's Theorem) is detailed. The distinction between a triple integral and an iterated integral in three variables is explained. Indeed, the triple integral is often evaluated by converting it to an equivalent iterated integral, which is usually easier to compute; but nonetheless triple integrals and iterated integration are distinct concepts. Essentially, an iterated integral is like the inverse of mixed partial differentiation and a triple integral is a direct extension of the Riemann integral (Riemann sums) of one and two variable to functions of three independent variables.

    We defined the definite integral of a single variable as a limit involving the Riemann sums



are points in a partition of the interval and is a representative point in the subinterval We can also apply the same idea to define a definite integral of two variables



over the rectangle First we partition the interval into subintervals and the interval into subintervals. Using these subdivisions, partition the rectangle into cells (subrectangles). Call this partition Choose a representative point from each cell in the partition of the rectangle. Form the sum,



where is the area of the -th representative cell. This is called the Riemann sum of with respect to the partition and the cell representation To measure the size of the rectangles in the partition we define the norm of the partition to be the length of the longest diagonal of any of the rectangle in the partition. We refine the partition by subdividing the cells in such a way that the norm decreases. When this process is applied to the Riemann sum and the norm decreases to zero, we write



If this limit exists, its value is called the double integral of over the rectangle
    We can now apply the same idea to define a definite integral of three variables



over a closed bound region in space.

Definition (Triple Integrals) Suppose   is defined on a closed  bounded solid region which in turn is contained in a box in space. We partition into a finite number of smaller boxes, call this partition   we choose a representative point from each subdivision in the partition and we form the sum,



where is the volume of the -th representative subdivision. This is called the Riemann sum of with respect to the partition and the cell representation To measure the size of the rectangles in the partition we define the norm of the partition to be the length of the longest diagonal of any of the subdivisions in the partition. We refine the partition by subdividing the subdivisions in such a way that the norm decreases. When this process is applied to the Riemann sum and the norm decreases to zero, we write



If this limit exists, its value is called the triple integral of over the closed bounded region

Proposition (Properties of Triple Integrals) Assume that all the given integrals exist on a rectangular region for given functions and

    (i)  Linearity Rule: For constants and
    

    
    (ii)   Dominance Rule:  If throughout a closed bounded region then
    


    (iii)   Subdivision Rule:  If the closed bounded region of integration is subdivided into two (disjoint) subdivisions and then

Proposition (Fubini's Theorem for Triple Integrals) If is continuous over a rectangular box : then the triple integral may be evaluated by the iterated integral



The iterated integral can be performed in any order, with appropriate adjustments to the limits of integration.

Example (Triple Integrals) Evaluate the triple integral where is the rectangular box

.

    Solution. We have
    









Example (Triple Integrals) Evaluate the triple integral where is the rectangular box

.

    Solution. We have
    

• print this page
• pages linking here
• related pages

Cite this as:
Triple Integrals
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/triple-integrals.html