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15.6withScannedExamples

# 15.6withScannedExamples - 15.6 TRIPLE INTEGRALS KIAM HEON G...

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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) €R3|ag :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 cry-plane. 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 sub-boxes 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 three-dimensional 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 right-hand 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 {By-plane. 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 R2|a_ < 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 R2|c_ < 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 yz-plane. 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 (3-3) 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 J "W147“? 52 7’0 ‘Tq9(%’»<)\(z’r0\6z)}=j ~ , ‘Igdkacvgygmpgv .__...._..___.... _.._._......_........———.—-_ WWW—.m-w.~—._.-.m.____ "g“ = Wag—.3 r. xp ti \$21 : 2 xPszﬁxx: if: . x00 Wpojﬂgﬁx] x3 [OJ 21%sz zQxCJXS 10f @ w .-._.-wo]d; 2%») an Q) (1/93 va [(Q ll J‘) ((0 10 lo) “HOV“- 100:}.m-f- }SIY‘ W P =23 349105+><w way Wﬁ- S‘DV‘ WWVVCEQ 344+ mam 4&3 ‘0?Z+ 9\—X Way) ”13 Wk} “.1. ”ZEN WP“: x. E: «f ...
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