alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

182 we need to be more clear about the terms change

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Unformatted text preview: tegration. In this chapter we cover a few examples to illustrate some of the techniques in integration. Let us review the 3 steps in integration. B B 5432 54321 54321 54321 54321 54321 54321 54321 1 54321 A 0 ∆xi x 0= a f(x) A b=xn x 0 x 0= a 1 1 1 1 1 1 1 1 1 1 1 f(x) δx i b=xn x APPROXIMATE step discrete summation discr 1. APPROXIMATE AREA step : discr ete summation of FINITELY many terms. n−1 F(x) = Σ i=0 n−1 ∆F(x i ) = Σ f (x i ). ∆x i . i=0 So : ∆F (xi ) = f(x i ). ∆x i is an element of area. . element step discrete summation 2. TENDS TO step : discrete summation as n → ∞ of COUNTABLY many terms. discr F(x) = n →∞ n →∞ Σ δF(x i ) = Σ f (x i ).. δx i i=0 i=0 So : δF (xi ) = f(x i ). δx i is an infinitesimal element of area . . infinitesimal step continuous summation 3. LIMIT step : continuous summation of UNCOUNTABLY many terms. contin n →∞ F(x) = Limit L imit b n →∞ b L imit Σ δ F(x i )=∫a dF(x)= Limit Σ f (xi) . dx = ∫ af (x).. dx i=0 i=0 Hence : dF(x) = f(x ). dx is an instantaneous element of area . instantaneous . 178 In the diagrams in the examples that follow we cannot depict an instantaneous instantaneous element f(x)dx . However, in keeping with the standard notation in text books we have used the label dx of the instantaneous element rather than δx of instantaneous the infinitesimal element . This should NOT give the wrong impression or infinitesimal infinitesimal interv wrong notion that dx is an infinitesimal inter v al . T he calculation with respect inf to dx be it differentiation or integration is always instantaneous . d ifferentiation i ntegration i nstantaneous infinitesimal From the diagrams the student should be able to construct an infinitesimal inf L i m i t f(x) δx . f(x) element , say f(x) δx , and then see that indeed f(x) dx = n → ∞ f(x) y ∆ = 2π/ n R } d arc = Rdθ 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 δθ = 2π/ n as n → ∞ dθ (0,0) θ R...
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