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Unformatted text preview: Lecture 8 3.2 Riemann Integral of Several Variables Last time we defined the Riemann integral for one variable, and today we generalize to many variables. Definition 3.3. A rectangle is a subset Q of R n of the form Q = [ a 1 , b 1 ] [ a n , b n ] , (3.10) where a i , b i R . Note that x = ( x 1 , . . . , x n ) Q a i x i b i for all i . The volume of the rectangle is v ( Q ) = ( b 1 a 1 ) n a n ) , (3.11) ( b and the width of the rectangle is width( Q ) = sup( b i a i ) . (3.12) i Recall (stated informally) that given [ a, b ] R , a finite subset P of [ a, b ] is a partition of [ a, b ] if a, b P and you can write P = { t i : i = 1 , . . . , N } , where t 1 = a < t 2 < . . . < t N = b . An interval I belongs to P if and only if I is one of the intervals [ t i , t i +1 ]. Definition 3.4. A partition P of Q is an ntuple ( P 1 , . . . , P n ), where each P i is a partition of [ a i , b i ]. Definition 3.5. A rectangle R = I 1 I n belongs to P if for each i , the interval I i belongs to P i . Let f : Q R be a bounded function, let P be a partition of Q , and let R be a rectangle belonging to P . We define m R f = inf f = g.l.b. { f ( x ) : x R } R (3.13) M R f = sup f = l.u.b. { f ( x ) : x R } , R from which we define the lower and upper Riemann sums, L ( f, P ) = m R ( f ) v ( R ) R (3.14) U ( f, P ) = M R ( f ) v ( R ) ....
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This note was uploaded on 10/02/2010 for the course MAT unknown taught by Professor Unknown during the Fall '04 term at MIT.
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