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Unformatted text preview: Chapter 7. Multiple Integrals This chapter is devoted to integrals of realvalued functions of two or three variables. Such integrals are called Multiple Integrals . 7.1. Double Integrals The definite integral R b a f ( x ) dx is defined with respect to a function f ( x ) defined over an interval a ≤ x ≤ b , and the result represents the (signed) area of the region between the graph of f and the interval a ≤ x ≤ b . The double integral Z Z D f ( x, y ) dx dy is defined with respect to a realvalued function f ( x, y ) of two variables defined over a closed and bounded region D in R 2 . The result represents the (signed) volume of the region between the surface defined by the graph of f and the region D : The definition of the double integral is similar to that of the definite integral. We first subdivide the region D into a large number of small rectangles by drawing parallels to the x and y axes. Number the small rectangles in D from 1 to n . Denote by A i the area of the i th rectangle. One then forms the sum n X i =1 f ( x i , y i ) A i where ( x i , y i ) T is an arbitary point in the i th rectangle. This sum approximates the volume under the surface. As the number of rectangles increases, the area of each individual rectangle shrinks to zero, and hence the double integral can be defined as the limit Z Z D f ( x, y ) dx dy = lim n →∞ n X i =1 f ( x i , y i ) A i provided that the limit exists. Iterated Integrals The direct evaluation of the above limit is generally very difficult. In practice, we shall com pute double integrals over a region by means of iterated integrals. We first study the case when the region is a rectangle and then more general regions later. Theorem 7.1 Suppose that f ( x, y ) is continuous on the rectangle R = [ a, b ] × [ c, d ] . Then Z Z R f ( x, y ) dx dy = Z b a Z d c f ( x, y ) dy ! dx = Z d c Z b a f ( x, y ) dx ! dy . Theorem 7.1 tells us how to determine a double integral by means of two successive single variable integrations. For the integral Z b a Z d c f ( x, y ) dy ! dx , we first hold x constant and integrate with respect to y from y = c to y = d . The result of the first integration Z d c f ( x, y ) dy is a function of x alone. Then we integrate this function with respect to x from x = a to x = b . Similarly, we can compute the iterated integral Z d c Z b a f ( x, y ) dx ! dy by first integrating from a to b with respect to x while keeping y fixed, and then integrating the result from c to d with respect to y . Theorem 7.1 guarantees that the value obtained is independent of the order of integration if f is continuous on R . Example 7.2 Compute in two different orders the double integral Z Z R 2 x 3 + 6 xy 2 dx dy over the rectangle R = [1 , 2] × [2 , 4] ....
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 Fall '11
 Wu
 Math, Integrals, dx dy, dr dθ dφ

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