Part 1: Definition of the Triple Integral

We can extend the concept of an integral into even higher dimensions. Indeed, in this section we develop the concept of a triple integral as an extension of the double integral definition.

To begin with, suppose that f(x,y,z) is a piecewise continuous function that assigns a number to each point in a solid W (the Greek capital "Omega" ). Further, suppose that f( x,y,z) is zero outside of  W and that W is contained within a parallelpiped

[ a,b] ×[ c,d]×[ p,q] = {( x,y,z)  | a £ x £ b, c £ y £ d, p £ z £ q}

(that is, [ a,b] ×[ c,d] ×[ p,q] is a "box").  

A Riemann sum of f( x,y,z) over the tagged partitions {x_j,t_j}_{j = 1}^m, { y_k,u_k}_{k = 1}ⁿ, and { z_l,v_l}_{l = 1}^r  of   [ a,b] , [c,d] , and [ p,q] , respectively, is a triple sum of the form

m å j = 1  n å k = 1  p å l = 1  f(tj,uk,vl) DxjDykDzl

The triple integral of f( x,y,z) over an arbitrary solid W is the limit as h approaches 0 of Riemann sums over h-fine partitions:

f( x,y,z) dV = lim h®0  m å j = 1  n å k = 1  o å l = 1  f(tj,uk,vl) DxjDykDzl

That is, the solid is "approximated" by a collection of "small boxes" with volume Dx_j Dy_k Dz_l .

 
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For example, if f( x,y) ³ g( x,y) over a region R in the xy-plane, then the triple integral of f( x,y,z) over the solid W bound between two surfaces z = g( x,y) and z = f( x,y) over the region R is given by

f( x,y,z) dV = é ë ó õ f( x,y) g(x,y)   f( x,y,z) dz ù û   dAxy

(1)

where dA_xy is the area differential in the xy-plane.        

EXAMPLE 1    Compute the triple integral of f(x,y,z) = 8xyz over the solid between z = 0 and z = 1 and over the region

R: x = 0 y = 2 x = 1 y = 3

Solution: To do so, we use (1) to write

8xyz  dV =   ó õ 1 0  8xyz  dz    dAxy =  4xyz2  1 0 dAxy = 4xy  dAxy

We then evaluate the resulting double integral over R:

8xyz  dV = ó õ 1 0  ó õ 3 2  4xy  dydx = 5

       

A similar derivation to that above shows that the volume of W is given by

Volume  of  W =  dV

(see the exercises).  Moreover, in analogy with (1), if  p(y,z) ³ q(y,z) over a region R in the yz-plane,

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Othen the triple integral of f(x,y,z) over the solid  W bound between the two surfaces x = q(y,z) and x = p( y,z) over the region R is given by

f( x,y,z) dV = é ë ó õ p( x,y) q(y,z)   f( x,y,z) dx ù û   dAyz

where dA_yz is the area differential in the yz-plane.

EXAMPLE 2    What is the volume of the solid between x = yz and x = 0 over the region y = 0, y = 1, z = 0, z = 4?       

Solution: The volume is given by

V =  dV = ó õ yz 0  dx  dAyz

Indeed, substituting the boundaries for R leads to the triple iterated integral

V = ó õ 4 0  ó õ 1 0  ó õ yz 0  dxdydz

Evaluating each integral in succession then leads to

V = ó õ 4 0  ó õ 1 0  yz  dydz = ó õ 4 0  z  y2 2 ê ê y = 1 y = 0  dz = ó õ 4 0   z 2 dz = 4