1873_solutions

is riemann integrable on 0 1 if and only if 13

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Unformatted text preview: se for a function f  f 1 , f 2 , f 3 defined on the rectangular box a 1 x b 1 and a 2 y b 2 and a 3 z b 3 . E  x, y, z This time we take 408 F x, y, z  Þa x 1 f 1 t, y, z dt  Þ y a2 f 2 a 1 , t, z dt  Þ z a3 f 3 a 1 , a 2 , t dt. Can you extend this idea to n dimensions? Some Exercises on Multiple Integrals 1. Evaluate the integral ÞÞ xy 2 dxdy S where S x, y x ÞÞ xy 2 dxdy  Þ 0 Þ 0 1 1x S 2. 0 and x  y 0 and y 1. xy 2 dydx  1 60 a. Convert the expression 4x 2 /9 Þ0 Þ0 3 f x, y dydx  Þ Þ 30 5 25 x 2 f x, y dydx into an integral of the form ÞÞ f where S is an appropriate subset of R 2 . You may assume that f is a S continuous function. The set S is R2 x, y 0 x 3 and 0 4x 2 9 y Þ x, y R2 3 x 5 and 0 y 25 x2 b. Convert the integral 4x 2 /9 Þ0 Þ0 3 f x, y dydx  Þ Þ 30 5 25 x 2 f x, y dydx into a repeated integral with order of integration reversed. Reversing the integral we obtain 4x 2 /9 Þ0 Þ0 3 f x, y dydx  Þ Þ 30 5 25 x 2 f x, y dydx  Þ0 Þ3 4 25 y 2 f x, y dxdy. y /2 3. Given that Q 3 is the standard simplex in R 3 defined by the equation R 3 x 0 and y 0 and z 0 and x  y  z Q 3  x, y, z 1, Q3 evaluate vol . The solution to this exercise was given in the preceding examples. 4. Given that S is the disc in R 2 with center at 0, 0 and radius r  0, show that the area vol S of S is given by the equation 409 vol S  Þ rÞ r r2 x2 r2 x2 1dydx  r 2 . The evaluation of this integral is a routine problem in elementary calculus. Note that the integral needs to be evaluated as a repeated integral. Students should be strongly discouraged from making a two-variable change to polar coordinates, even if they learned such a procedure in their elementary calculus courses. That procedure will not be available till much later in this chapter. 5. Given that A and B are subsets of R n and R k respectively where n and k are positive integers and given that the volumes vol A and vol B exist, prove that the measure vol A B exists in the space R nk and that vol A B  vol A vol B . Suppose that E is a nk cell that includes the set A B. We can express E in the form E 1 E 2 where E 1 is an n-cell that includes A and E 2 is a k-cell that includes B. Then vol A B ÞE  A B  ÞE 1 E2 ÞE AB  A 1 ÞE B  vol A vol B 2 6. Suppose that for each A n is an expanding sequence of subsets of R k , that A Ý  An n1 and that the set A is bounded. Suppose that each of the sets A n is a finite union of k-cells. Prove that vol A  n Ý vol A n . lim We choose a k-cell E that includes A. Since the repeated integral of  A n over E exists for each n and since  A n converges boundedly to  A as n Ý, we have vol A  lim ÞE  A  n Ý ÞE n  A n  n Ý vol A n . lim 7. Suppose that for each A n is an contracting sequence of subsets of R k , that A Ý  An n1 and that the set A 1 is bounded. Suppose that each of the sets A n is a finite union of k-cells. Prove that vol A  n Ý vol A n . lim Choose a k-cell E that includes A 1 . Since the s...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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