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Unformatted text preview: Lecture XIV Multiple Integrals 1 Integrals of one-variable functions For a real-valued function f defined on an interval [ a, b ], the integral of f over b [ a, b ], denoted by a fdx , is defined as follows: n b fdx = lim f ( x i ) x i , n , max x i a i =1 where a = x 1 &lt; . . . &lt; x i &lt; x i +1 &lt; x n = b , x i = x i x i 1 and &lt; x i [ x i 1 , x i ] for all i 2 , i n . We will now give a few properties and theorems about integrals in one-variable calculus. Theorem 1 (First Existence Theorem) Let f be a one-variable function. b If f is continuous on [ a, b ] , then a fdx exists. b b Let f and g be two functions defined on [ a, b ] such that a fdx and gdx exist. a Then the following properties hold: 1. Let c be such that a &lt; c &lt; b . Then the following integrals exist and the equality holds: b c b fdx = fdx + fdx. a a c 2. The integral in the left side of the equality exists and the equality holds: b b b ( f + g ) dx = fdx + gdx. a a a 3. If c is a constant, the integral in the left side of the equality exists and the equality holds: b b ( cf ) dx = c fdx. a a 1 2 4. If m 1 and m 2 are two constants such that m 1 f ( x ) m 2 for all x [ a, b ], then b m 1 ( b a ) fdx m 2 ( b a ) . a Scalar-valued integrals in E 2 and E 3 If f is a scalar-valued function defined on E 2 or E 3 , how do we define the integrals of f on a region R ? We can only do this on a special kind of region, called regular , which we will define in section 5. Definition 1 Let R be a regular region in E 2 and f a scalar field on R . The integral of f on R , denoted by R fdA , is defined as follows: n fdA = lim f ( P i ) A i , R n max d i 0 i =1 where A 1 , . . . , A n form a subdivision of R into elementary regions, A i is the area of A i , P i is a point in A i and d i is the diameter of A i , i.e. , i....
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