Chapter 10 Multivariable Integral
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Chapter 10 Multivariable integral 10.1 Riemann integral over rectangles Note: ??? lectures As in chapter chapter 5, we define the Riemann integral using the Darboux upper and lower inte- grals. The ideas in this section are very similar to integration in one dimension. The complication is mostly notational. The differences between one and several dimensions will grow more pronounced in the sections following. 10.1.1 Rectangles and partitions Definition 10.1.1. Let (a ,a ,...,a ) and (b ,b ,...,b ) be such that a b for all k. A set of 1 2 n 1 2 n k ≤ k the form [a ,b ] [a ,b ] [a ,b ] is called a closed rectangle. In this setting it is sometimes 1 1 × 2 2 ×···× n n useful to allow a =b , in which case we think of [a ,b ] = a as usual. If a <b for all k, then k k k k { k} k k a set of the form(a 1,b 1) (a 2,b 2) (a n,b n) is called an open rectangle. × ×···× n For an open or closed rectangle R:= [a 1,b 1] [a 2,b 2] [a n,b n] R or R:= (a 1,b 1) n × ×···× ⊂ × (a2,b 2) (a n,b n) R , we define the n-dimensional volume by ×···× ⊂ V(R):= (b a )(b a ) (b a ). 1 − 1 2 − 2 ··· n − n A partition P of the closed rectangle R=[a ,b ] [a ,b ] [a ,b ] is afinite set of par- 1 1 × 2 2 ×···× n n titions P1,P2,...,Pn of the intervals [a1,b 1],[a 2,b 2],...,[a n,b n]. We will write P=(P1,P2,...,Pn). That is, for every k there is an integer � and thefinite set of numbers P = x ,x ,x ,...,x k k { k,0 k,1 k,2 k,�k } such that ak =x k,0 <x k,1 <x k,2 < <x k,� 1 <x k,� =b k. ··· k− k Picking a set ofn integersj ,j ,...,j wherej 1,2,...,� we get the subrectangle 1 2 n k ∈{ k} [x1,j 1 ,x 1,j ] [x 2,j 1 ,x 2,j ] [x n,j 1 ,x n,j ]. 1− 1 × 2− 2 ×···× n− n 75 76 CHAPTER 10. MULTIVARIABLE INTEGRAL For simplicity, we order the subrectangles somehow and we say R ,R ,...,R are the subrectan- { 1 2 N} gles corresponding to the partition P of R. Or more simply, we say they are the subrectangles of P. In other words we subdivide the original rectangle into many smaller subrectangles. It is not difficult to see that these subrectangles cover our original R, and their volume sums to that of R. That is N N R= R j, andV(R) = ∑ V(R j). j=1 j�=1 When Rk = [x1,j 1 ,x 1,j ] [x 2,j 1 ,x 2,j ] [x n,j 1 ,x n,j ] 1− 1 × 2− 2 ×···× n− n then V(R k) =Δx 1,j Δx2,j Δx n,j = (x1,j x 1,j 1)(x2,j x 2,j 1) (x n,j x n,j 1). 1 2 ··· n 1 − 1− 2 − 2− ··· n − n− Let R R n be a closed rectangle and let f:R R be a bounded function. Let P be a partition ⊂ → of[a,b] and suppose that there areN subrectangles. LetR i be a subrectangle ofP. Define m := inf f(x):x R , i { ∈ i} M := sup f(x):x R , i { ∈ i} N L(P,f):= ∑ miV(R i), i=1 N U(P,f):= ∑ MiV(R i). i=1 We callL(P,f) the lower Darboux sum andU(P,f) the upper Darboux sum. The indexing in the definition may be complicated, fortunately we generally do not need to go back directly to the definition often. We start proving facts about the Darboux sums analogous to the one-variable results. Proposition 10.1.2. Suppose R R n is a closed rectangle and f:R R is a bounded function. ⊂ → Let m,M R be such that for all x R we have m f(x) M. For any partition P of R we have ∈ ∈ ≤ ≤ mV(R) L(P,f) U(P,f) MV(R). ≤ ≤ ≤ Proof. Let P be a partition. Then note that m m i for all i and Mi M for all i. Also mi M i for N ≤ ≤ ≤ alli. Finally ∑i=1 V(R i) =V(R). Therefore, N N N mV(R) =m ∑ V(R i) = ∑ mV(R i) ∑ miV(R i) �i=1 � i=1 ≤ i=1 ≤ N N N ∑ MiV(R i) ∑ MV(R i) =M ∑ V(R i) =MV(R). ≤ i=1 ≤ i=1 �i=1 � 10.1. RIEMANN INTEGRAL OVER RECTANGLES 77 10.1.2 Upper and lower integrals By Proposition 10.1.2 the set of upper and lower Darboux sums are bounded sets and we can take their infima and suprema. As before, we now make the following definition. Definition 10.1.3. Iff:R R is a bounded function on a closed rectangleR R n. Define → ⊂ f:= sup L(P,f):P a partition ofR , f:= inf U(P,f):P a partition ofR . { } { } �R �R We call the lower Darboux integral and the upper Darboux integral. As in� one dimension we have refinements� of partitions. n Definition 10.1.4. Let R R be a closed rectangle and let P=(P1,P2,...,Pn) and P=( P1,P2,...,Pn) ⊂ be partitions ofR. We say Pa refinement ofP if as setsP P for allk=1,2,...,n. k ⊂ k � � � � It is not difficult to see that if P is a refinement of P, then subrectangles of P are unions of � � subrectangles of P. Simply put, in a refinement we took the subrectangles of P and we cut them into smaller subrectangles. � � Proposition 10.1.5. Suppose R R n is a closed rectangle, P is a partition of R and P is a refinement ⊂ of P. If f:R R be a bounded function, then → � L(P,f) L( P,f) and U( P,f) U(P,f). ≤ ≤ Proof. Let R1,R 2,...,R N be the subrectangles� of P and R�1,R2,...,RM be the subrectangles of R. LetI be the set of indicesj such that R R . We notice that k j ⊂ k � � � � Rk = �R j,V(R k) = ∑ V(R j). j I j Ik �∈ k ∈ � � Let m := inf f(x):x R , and m := inf f(x): R as usual. Notice also that if j I , then j { ∈ j} j { ∈ j} ∈ k m m . Then k ≤ j � N N � N M � L(P,f)= ∑ mkV(R k) = ∑ ∑ mkV(R j) ∑ ∑ m jV(R j) = ∑ m jV(R j) =L( P,f). k=1 k=1 j I ≤ k=1 j I j=1 ∈ k ∈ k The key point of this next proposition is� that the lower� Darboux� integral� is less� than or� equal to the upper Darboux integral. Proposition 10.1.6. Let R R n be a closed rectangle and f:R R a bounded function. Let ⊂ → m,M R be such that for all x R we have m f(x) M. Then ∈ ∈ ≤ ≤ mV(R) f f MV(R). (10.1) ≤ ≤ ≤ �R �R 78 CHAPTER 10. MULTIVARIABLE INTEGRAL Proof. For any partitionP, via Proposition 10.1.2 mV(R) L(P,f) U(P,f) MV(R). ≤ ≤ ≤ By taking suprema of L(P,f) and infima of U(P,f) over all P we obtain thefirst and the last inequality. The key of course is the middle inequality in (10.1). Let P=(P1,P2,...,Pn) and Q=(Q 1,Q 2,...,Q n) be partitions of R. Define P=( P ,P ,...,P ) by letting P =P Q . Then P is a partition of R as 1 2 n k k ∪ k can easily be checked, and P is a refinement of P and a refinement of Q. By Proposition 10.1.5, L(P,f) L( P,f) andU( P�,f) U�(Q�,f). Therefore,� � � ≤ ≤ � L(P,f) L( P,f) U( P,f) U(Q,f). � � ≤ ≤ ≤ In other words, for two arbitrary partitions P and Q we have L(P,f) U(Q,f) . Via Proposition ?? � � ≤ we obtain sup L(P,f):P a partition ofR inf U(P,f):P a partition ofR . { }≤ { } In other words f f. R ≤ R � � 10.1.3 The Riemann integral We now have all we need to define the Riemann integral in n-dimensions over rectangles. Again, the Riemann integral is only defined on a certain class of functions, called the Riemann integrable functions. Definition 10.1.7. Let R R n be a closed rectangle. Let f:R R be a bounded function such ⊂ → that f(x)dx= f(x)dx. �R �R Then f is said to be Riemann integrable. The set of Riemann integrable functions on R is denoted byR(R). Whenf R(R) we define the Riemann integral ∈ f:= f= f. �R �R �R When the variablex R n needs to be emphasized we write ∈ f(x)dx, f(x ,...,x )dx dx , or f(x) dV. 1 n 1 ··· n �R �R �R IfR R 2, then often instead of volume we say area, and hence write ⊂ f(x)dA. �R Proposition 10.1.6 implies immediately the following proposition. 10.1. RIEMANN INTEGRAL OVER RECTANGLES 79 Proposition 10.1.8. Let f:R R be a Riemann integrable function on a closed rectangle R R n.