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Unformatted text preview: DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II FURTHER APPLICATIONS OF INTEGRATION Chapter 4 DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II Suppose that a curve C is defined by the equation y=f(x), where f is continuous and axb. We approximate C by a polygonal which is obtained by dividing the interval [a,b] into n equal subintervals with endpoints 1 n x ,x , ,x L I . ARC LENGTH DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II Approximation by a polygon DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II and width x. If , then the point The polygon with vertices is an approximation to C. The length L of C is approximately the length of this polygon and the approximation gets better i i y f (x ) = ( , ) i i i P x y C P 1 , , n P P P L DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II as we let n increase. We define the length L of the curve C with equation y=f(x), axb is the limit 1 1 lim n i i n i L P P = = DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II We let then we have By applying the Mean Value Theorem to f on the interval , there exists a number between such that 1 , i i i y y y = 2 2 2 2 1 1 1 ( ) ( ) ( ) ( ) i i i i i i i P P x x y y x y = + = + 1 [ , ] i i y y * i x 1 and i i x x * 1 1 ( ) ( ) '( )( ) i i i i i f x f x f x x x = DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II That is Thus, we have * '( ) i i y f x x = 2 2 1 2 2 * 2 * ( ) ( ) ( ) '( ) 1 '( ) i i i i i P P x y x f x x f x x = + = + = + DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II Therefore, According to the definition of the definite integrals, we see that the right hand side is equal to 2 * 1 1 1 lim lim 1 '( ) n n i i i n n i i L P P f x x = = = = + [ ] 2 1 '( ) b a f x dx + DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II DANANG UNI VERSI TY OF TECH NOLOGY CALCULUS WITH ANALYTIC GEOMETRY II The Arc Length Formula If f is continuous on [a,b], then the length of the...
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 Spring '09
 Dinh

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