3. Triple Integrals
3.1. Physical Meaning of Double Integrals
Consider a lamina occupying a region D⊆R2with density function ρ(x, y). Then its total mass is
ZZD
ρ(x, y)dA.
Indeed, assume for simplicity that Dis a rectangle. We can split it into small subrectangles
R1, . . . , Rn, and choose a point (xi, yi)∈Riin each subrectangle. On each subrectangle Ri, the
density ρ(x, y) is almost constant: ρ(x, y)≈ρ(xi, yi). The mass of Riis approximately its area times
this almost constant density: ρ(xi, yi) Area(Ri). Thus, the total mass is approximately
n
X
i=1
ρ(xi, yi) Area(Ri)≈ZZD
ρ(x, y)dA.
3.2. Triple Integrals over Boxes
They are defined and calculated in much the same way as double integrals.
First, assume Bis a box, i.e. a region of the type
B= [a, b]×[c, d]×[p, q] = {a≤x≤b, c ≤y≤d, p ≤z≤q}.
Assume a function f:B→Ris defined on this box. Let us split it into small sub-boxes
B1, B2, . . . , Bn. Pick a point (x1, y1, z1)∈B1,(x2, y2, z2)∈B2, . . . in each of these small boxes.
Then
ZZZB
fdV = lim
n
X
i=1
f(xi, yi, zi) Vol(Bi),
where the limit is taken as the sizes of boxes tend to zero. We can calculate it as an interated integral:
ZZZB
fdV =Zq
pZd
cZb
a
f(x, y, z)dxdy dz
We can take any order of integration, say first y, then z, then x. The result will be the same.
You cannot visualize it as the region under the graph of f, this would require 4D space. However,
you can visualize it as a mass of lamina occupying the region Bwith density ρ(x, y, z) at the point
(x, y, z). This is similar to a double integral as the mass of a lamina.
3.3. Example 1
Evaluate RRRByzdV , where B= [0,1] ×[0,1] ×[0,1]. We have:
ZZZB
yzdV =Z1
0Z1
0Z1
0
yzdz dydx =Z1
0"Z1
0
yz2
2
z=1
z=0
dy#dx =
Z1
0Z1
0
y
2dydx =Z1
0
y2
4
y=1
y=0
dx =Z1
0
1
4dx =x
4
x=1
x=0 =1
4.
1
