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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 twovariable 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 ncell that includes A and E 2 is a kcell 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 kcells. Prove that
vol A n Ý vol A n .
lim
We choose a kcell 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 kcells. Prove that
vol A n Ý vol A n .
lim
Choose a kcell 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.
 Fall '08
 STAFF
 Math, Calculus

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