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chapter8 - Chapter 8 Multiple Integrals 8.1 Double...

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Unformatted text preview: Chapter 8. Multiple Integrals 8.1 Double Integrals 8.1.1 Definition The definition of definite integral in chapter 3 can be extended to functions of two variables. Let R be a plane region in the xy-plane. Subdivide R into subrectangles R i ( i = 1 ,...,n ) . Let Δ A i be the area of R i and ( x i ,y i ) be a point in R i . Let f ( x,y ) be a function of two variables. Then the double integral of f over R is ZZ R f ( x,y ) dA = lim n →∞ n X i =1 f ( x i ,y i )Δ A i . 2 MA1505 Chapter 8. Multiple Integrals 8.1.2 Geometrical meaning Geometrically, if f ( x,y ) ≥ 0 for all ( x,y ) ∈ R, the definite integral RR R f ( x,y ) dA is equal to the vol- ume under the surface z = f ( x,y ) and above the xy-plane over the region R as shown in the following diagram. y z z=f(x,y) a b x c d R Summing over all the rectangles, n X i =1 f ( x i ,y i ) A i gives the approximate volume of the solid under the 2 3 MA1505 Chapter 8. Multiple Integrals surface and above R . By letting n go to ∞ , (i.e. making the subdivision more refined), the above sum will approach the exact volume of the solid. 8.1.3 Properties of Double Integrals (1) ZZ R ( f ( x,y ) + g ( x,y )) dA = ZZ R f ( x,y ) dA + ZZ R g ( x,y ) dA. (2) ZZ R cf ( x,y ) dA = c ZZ R f ( x,y ) dA, where c is a constant. (3) If f ( x,y ) ≥ g ( x,y ) for all ( x,y ) ∈ R , then ZZ R f ( x,y ) dA ≥ ZZ R g ( x,y ) dA. (4) ZZ R dA = ZZ R 1 dA ¶ = A ( R ), the area of R . 3 4 MA1505 Chapter 8. Multiple Integrals (5) ZZ R f ( x,y ) dA = ZZ R 1 f ( x,y ) dA + ZZ R 2 f ( x,y ) dA , where R = R 1 ∪ R 2 and R 1 , R 2 do not overlap except perhaps on their boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................................ ............................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R 1 R 2 R (6) If m ≤ f ( x,y ) ≤ M for all ( x,y ) ∈ R , then mA ( R ) ≤ ZZ R f ( x,y ) dA ≤ MA ( R ). 8.2 Evaluation of double integrals We shall discuss how to derive an efficient way to evaluate double integrals over certain plane regions. The key is to describe the given region in terms of the coordinates. 4 5 MA1505 Chapter 8. Multiple Integrals 8.2.1 Rectangular regions A rectangular region R in the xy-plane can be de- scribed in terms of inequalities: a ≤ x ≤ b, c ≤ y ≤ d. Then ZZ R f ( x,y ) dA = Z d c " Z b a f ( x,y ) dx # dy. The RHS is called an iterated integral . i.e. re- peating the integration for each variable, one at a time. We can also change the order of the variables of in- tegration (without changing the value): ZZ R f ( x,y ) dA = Z b a " Z d c f ( x,y ) dy # dx....
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