Section 16.7 Triple Integrals. TRIPLE INTEGRAL OVER A BOX Consider a function w = f (x, y, z) of...
21
Section 16.7 Triple Integrals
-
Upload
allen-quinn -
Category
Documents
-
view
225 -
download
0
Transcript of Section 16.7 Triple Integrals. TRIPLE INTEGRAL OVER A BOX Consider a function w = f (x, y, z) of...
Section 16.7
Triple Integrals
TRIPLE INTEGRAL OVER A BOX
Consider a function w = f (x, y, z) of three variables defined on the box B given by
Divide B into sub-boxes by dividing the interval [a, b] into l subintervals of equal width Δx, dividing [c, d] into m subintervals of equal width Δy, and dividing [r, s] into n subintervals of equal width Δz. This divides the box B into l∙m∙n sub-boxes. A typical sub-box Bilk is
Bijk = [xi − 1, xi] × [yj − 1, yj] × [zk − 1, zk]
Each sub-box has volume ΔV = Δx Δy Δz.
],[],[],[
},,|),,{(
srdcba
szrdycbxazyxB
We now form the triple Riemann sum
where the sample point is in box Bijk.
TRIPLE INTEGRAL OVER A BOX (CONTINUED)
l
i
m
j
n
kijkijkijk Vzyxf
1 1 1
*** ),,(
),,( ***ijkijkijk zyx
The triple integral of f over the box B is
if this limit exists.
The triple integral always exists if f is continuous. If we choose the sample point to be (xi, yj, zk), the triple integral simplifies to
TRIPLE INTEGRAL OVER A BOX (CONCLUDED)
l
i
m
j
n
kijkijkijk
Bnml
VzyxfdVzyxf1 1 1
***
,,),,(lim),,(
l
i
m
j
n
kkji
Bnml
VzyxfdVzyxf1 1 1
,,),,(lim),,(
FUBINI’S THEOREM FOR TRIPLE INTEGRAL
Theorem: If f is continuous on the rectangular box B = [a, b] × [c, d] × [r, s], then
s
r
d
c
b
aB
dzdydxzyxfdVzyxf ),,(),,(
NOTE: The order of the partial antiderivatives does not matter as long as the endpoints correspond to the proper variable.
EXAMPLE
Evaluate the triple integral , where
B is the rectangular box given by
B = {(x, y, z) | 1 ≤ x ≤ 2, 0 ≤ y ≤ 1, 0 ≤ z ≤ 2}
B
dVzyx2
TRIPLE INTEGRAL OVER A BOUNDED REGION
The triple integral over the bounded region E is defined as
where B is a box containing the region E and the function F is defined as
BE
dVzyxFdVzyxf ),,(),,(
EBzyx
EzyxzyxfzyxF
innotbutis),,(if0
inis),,(if),,(),,(
The region E is said to by of type 1 if it lies between to continuous functions of x and y. That is,
where D is the projection of E onto the xy-plane.
The triple integral over a type 1 region is
TYPE 1 REGIONS
D
yxu
yxuE
dAdzzyxfdVzyxf).(
),(
2
1
),,(),,(
)},(),(,),(|),,{( 21 yxuzyxuDyxzyxE
If D is a type I region in the xy-plane, then E can be described as
and the triple integral becomes
TYPE 1 REGIONS (CONTINUED)
)},(),(),()(,|),,{( 2121 yxuzyxuxgyxgbxazyxE
E
b
a
xg
xg
yxu
yxudxdydzzyxfdVzyxf
)(
)(
),.(
),(
2
1
2
1
),,(),,(
If D is a type II region in the xy-plane, then E can be described as
and the triple integral becomes
TYPE 1 REGIONS (CONCLUDED)
)},(),(),()(,|),,{( 2121 yxuzyxuyhxyhdyczyxE
E
d
c
yh
yh
yxu
yxudydxdzzyxfdVzyxf
)(
)(
),.(
),(
2
1
2
1
),,(),,(
EXAMPLE
Evaluate the triple integral , where
E is the region bounded by the planes x = 0, y = 0, z = 0, and 2x + 2y + z = 4.
E
dVy
The region E is said to by of type 2 if it lies between two continuous functions of y and z. That is,
where D is the projection of E onto the yz-plane.
The triple integral over a type 2 region is
TYPE 2 REGIONS
D
zyu
zyuE
dAdxzyxfdVzyxf),(
),(
2
1
),,(),,(
)},(),(,),(|),,{( 21 zyuxzyuDzyzyxE
The region E is said to by of type 3 if it lies between two continuous functions of x and z. That is,
where D is the projection of E onto the xz-plane.
The triple integral over a type 3 region is
TYPE 3 REGIONS
D
zxu
zxuE
dAdyzyxfdVzyxf).(
),(
2
1
),,(),,(
)},(),(,),(|),,{( 21 zxuyzxuDzxzyxE
EXAMPLE
Evaluate the triple integral ,
where E is the region bounded by the paraboloid x = y2 + z2 and the plane x = 4.
E
dVzy 22
VOLUME AND TRIPLE INTEGRALS
The triple integral of the function f (x, y, z) = 1 over the region E gives the volume of E; that is,
E
dVEV 1)(
EXAMPLE
Find the volume of the region E bounded by the plane z = 0, the plane z = x, and the cylinder x = 4 − y2.
MASS
Suppose the density function of a solid object that occupies the region E is ρ(x, y, z). Then the mass of the solid is
E
dVzyxm ),,(
MOMENTS
Suppose the density function of a solid object that occupies the region E is ρ(x, y, z). Then the moments of the solid about the three coordinate planes are
E
xy
E
xz
E
yz
dVzyxzM
dVzyxyMdVzyxxM
),,(
),,(),,(
CENTER OF MASS
),,( zyxThe center of mass is located at the point
where
m
Mz
m
My
m
Mx xyxzyz
If the density is constant, the center of mass of the solid is called the centroid of E.
MOMENTS OF INERTIA
The moments of inertia about the three coordinate axis are
E
z
E
y
E
x
dVzyxyxI
dVzyxzxI
dVzyxzyI
),,()(
),,()(
),,()(
22
22
22
EXAMPLE
Find the mass and center of mass of the tetrahedron bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 1 whose density function is given by ρ(x, y, z) = y.
