Multivariate Calculus - Bruce E. Shapiro

LECTURE 20. TRIPLE INTEGRALS 163
Finally, we an integrate over z.
∫ π/2 ∫ z ∫ y
0
0
0
sin(x + y + z)dxdydz =
∫ π/2
0
[− 1 2 sin 3z + sin 2z − 1 2 sin z ]
dz
[ 1
=
6 cos 3z − 1 2 cos 2z + 1 ] π/2
2 cos z 0
[ 1
=
6 cos 3π 2 − 1 2 cos π + 1 2 cos π ]
2
[ 1
−
6 cos 0 − 1 2 cos 0 + 1 ]
2 cos 0
= 1 6 (0) − 1 2 (−1) + 1 2 (0) − 1 6 = 1 2
Definition 20.1 If S ∪ R 3 is any solid object and f(x, y, z) : S ↦→ R then the
volume integral of f over S
I = f(x, y, z)dxdydz
s
Theorem 20.1 The volume of S is the volume integral of f(x, y, z) = 1 over S
V = dxdydz
s
The volume of S is the volume integral of the function f (x,y,z)=1.
Example 20.3 . Find the volume of the solid in the first octant bounded by the
surfaces
y 2 + 64z 2 = 4
and
y = x
Figure 20.1: Left: the portion of the elliptic cylinder y 2 +64z 2 = 4 in the first octant
. Right: the portion bounded by y = x in the first octant.
Solution. We can solve the equation of the surface for z,
z = ± 1 8√
4 − y 2
Math 250, Fall 2006 Revised December 6, 2006.