section 16.7
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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Section 16.7
Triple Integrals
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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
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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
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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),,(
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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.
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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
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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),,(),,(
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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
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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
),,(),,(
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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
),,(),,(
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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
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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
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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
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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
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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)(
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EXAMPLE
Find the volume of the region E bounded by the plane z = 0, the plane z = x, and the cylinder x = 4 − y2.
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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 ),,(
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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
),,(
),,(),,(
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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.
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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
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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.