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Unformatted text preview: 15.6: TRIPLE INTEGRALS KIAM HEON G KWA 1. TRIPLE INTEGRALS OVER A RECTANGULAR Box Let f (as, y, .2) be a function on a rectangular box
B = [a,b] >< [c,d] X [738]
= {(cc,y,z) €R3ag :6 S b,c§y S d,7" 3 z S s}, where—oo<a§b<oo,—oo<c£d<oo,and—oo<r§s<oo.
The triple integral of f over B is deﬁned in a way very similar to the
double integral of a function of two variables over a rectangular region
in the cryplane. First we divide the intervals [0,, b], [c, d], and [733]
into I subintervals [mi—1,11%] of width A3; = (b — a)/l, m subintervals
[yj_1, yj] of width Ay = (d—c)/m, and n subintervals [zk_1, 2k] of width
A2 = (s—r)/n, respectively, where i = 1, 2,  ~ ,l, j = 1, 2,   ,m, and
k = 1,2,  .. ,n. This process results in the decomposition of B into
lmn subboxes Bijk = [xi—1,11%] X [yj—layj] X [Zk—1,Zkl, each of which has a volume of AV = AxAyAz. Then by selecting from
each Bijk a sample point (my, 31;, z,’;), we form a triple Riemann sum m n l
ZZZf(x:,y;,z;)Av, i=1 j=1 k=1 and deﬁne the triple integral of f over B as provided the limit exists. As in the case of double integrals, if f is continuous on B, then
(1.2) s d b
Fubini’s theorem: /// f(a:,y,z)dV = / / / f(:r,y,z) d3: dy dz.
B r c a Date: October 15, 2010. 2 KIAM HEONG KWA It should be noted that there are ﬁve other possible orders of integra
tion, among which //Bf(m,y,z)dV=/Ts/ab/cd f(m,y,z)dyd$dz is such a possible order. 2. TRIPLE INTEGRALS OVER A GENERAL BOUNDED REGION As in the case of double integrals, if f is given on a bounded solid
region E in the threedimensional Euclidean space, we deﬁne its triple
integral over E by setting (2.1) ///E f(w,y,z)dV=///B F(m,y,z)dV where B is a rectangular box enclosing E and F is the function that
agrees with f on E but is O for points in B but not in E, provided
the integral on the righthand side of (2.1) exists. The integral in (2.1)
does not exist without further conditions on f as well as on E. We
give only a very restrictive set of conditions. In the sequel, we assume that f is continuous. A solid region E is said to be of type 1 if it is bounded below and above by the graphs of two continuous functions u1(x, y) and u2(a;, y)
deﬁned over a bounded region D in the {Byplane. That is to say, E = {(ac,y,z) e 1R3(a:,y)€ D, u1(z,y) s z s u2(x,y)} In this case, we have // W >=dV //l/. Furthermore, if D is a plane region of type I, so that D = {(ac,y) E
R2a_ < a: < b ,ccgl( ) < y < 92(56 )}( for some scalars a and b and some
continuous functions 91(55) and 92(x ,we have 92(26) U20”, y)f
(2.3) ///E f( x, y,z) dV= /b /) / f(,x y,z) dzdydx.
91(1) U1(w y) u2($ 10f
f(m ,y,z) dz dA 1(xyy) 15.6: TRIPLE INTEGRALS 3 Similarly, if D is a plane region of type II, so that D = {(x,y) E
R2c_ < y__ < d, h1(y ) < ac < h2(y )} for some scalars c and d and some continuous functions h1(y) and h2( (,y) we have h2(y) u2(a:,y)
®/// f x y~=ZW / / / f<x,y,z)dzdxdy.
h1(y) 1L1(:1:,y) A solid region E is said to be of type 2 if it is bounded from the left
and from the right in the m—direction by the graphs of two continuous
functions 111(y, z) and 122(y,z) deﬁned over a bounded region D in the
yzplane. That is to say, E = {(x,y, z) E R3(y,z) E D,v1(y,z)g a: S 02(y,z)}. In this case, we have (2.5) ///E f(:L‘,y,z)dV= //D [/U::::) f(a:,y,z)da: dA. Finally, the solid region E is said to be of type 3 if it is bounded
from the left and from the right in the y—direction by the graphs of
two continuous functions w1(a:, 2) and 1112 (3:, 2) deﬁned over a bounded
region D in the xz—plane. That is to say, E = {($,y,z) E R3(m,z) E D, w1(x, 2) S y S w2(:c,z)}. In this case, we have
102 (z, z)f
/ f(ac y,,z) dy dA.
w1(z,z) /// “w W =// l 3. SOME APPLICATIONS OF TRIPLE INTEGRALS (2.6) Volume of a bounded region. If E is a bounded solid region, then
its volume is given by (3.1) V(E) = //E dV provided the integral exists. Let p($,y,z) be the mass density function of a solid object that
occupies the region E in the sequel. 4 KIAM HEONG KWA Mass of a solid. The mass of the solid is given by (3.2) m = ///E p(a:,y,z)dV provided the integral exists. Moments about coordinate planes. The moments of the solid
about the yz—, 372—, and my—planes are (33) Myz= ///E wp(w,y,z)dV,
sz=///E yp($,y,Z)dV,
Mm: ///E zp<w,y,z)dv, respectively, provided the integrals exist. Center of mass. The center of mass of the solid is the point (3‘3, 37, 2), Where
(3.4) at = M?” ‘ Mzz Macy
a y =
m ,and2= .
m If the mass density is constant throughout the solid, the center of mass
is also called the centroid. Moments of inersia about coordinate axes. The moments of in
ersia about the 33—, y—, and 2—, axes are (35) Ian = ///E (y2 +22)p(x,y,z)dV,
Iy = ///E($2 + 22)p(:r,y,z)dV,
I. = #4 (x2 + mm, .11. z) dv, respectively, provided the integrals exists. T
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 Fall '10
 Kwa
 Calculus, Geometry

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