7-Multiple Integrals_1407522557.pdf - 7 Multiple integrals...

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1 7. Multiple integrals 7.1 Double integrals over Rectangles Recall the definition of definite integral of function with one variable. Let f: [a, b] → , a < b, a, b . Choose an integer n >0. Divide the interval [a, b] into n equal subintervals. Note that [a, b] has length b – a, so each of the subintervals has length b a x n = .Thus, , , Choose a sample point x i * [x i-1 , x i ], and form the Riemann Sum 1 2 1 ( *) ( *) ... ( *) ( *) n n i i f x x f x x f x x f x x = ∆ + ∆ + + = Notice that f(x i *) Δx is the area of the rectangle with base [x i-1 , x i ] and height f(x i *). The integral of f on [a, b] is 1 ( ) ( ) lim n b i a n i f x dx f x x →∞ = = . If f ≥ 0, ( ) b a f x dx is the area of the region above [a, b] under the graph of f.
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