alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

Definite b b fxdx fx fxdx a a a fxdx fxdx b

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Unformatted text preview: 321 4321 4321 4321 4321 4321 4321 4321 A x x 0= a ∆xi b=xn (0 , 0 ) APPROXIMATE step discrete summation discr 1. APPR OXIMATE AREA step : discr ete summation of FINITELY many terms. n−1 An approximation of the actual area function F(x) = Σ ∆F(x i ) i=0 = f(x0) (x1 -- a) + f (x1) (x2 -- x1) + ... + f(xn -- 1) (b -- xn -- 1) = --where ∆F (xi ) = f(x i ). ∆x i an element of area . . element n−1 Σ f (x i ).. ∆x i i=0 step discrete summation 2. TENDS TO step : discrete summation as n → ∞ of COUNTABLY many terms. discr n →∞ n →∞ i=0 i=0 Σ δF(x i ) = f(x0) δ x0 + f (x1) δx1 + . . . + f(x n -- 1) δx n -- 1= Σ f (x i ). δ x i . where δF (xi ) = f(x i ). δx i an infinitesimal element of area . . infinitesimal step continuous summation contin 3. LIMIT step : continuous summation of UNCOUNTABLY many terms. n →∞ F(x) = Limit L imit Σ i=0 b δ F(x i )=∫ dF(x)= Limit L imit a n →∞ b Σ f(xi) . dx = ∫ af (x).. dx i=0 . where dF(x) = f(x ). dx an instantaneous element of area . instantaneous So f(x) = d F(x) . The ar ea function F(x) = ANTIDERIVATIVE {f(x)}. area ANTIDERIVA dx 137 b F(x ∫ f(x)dx = CHANGE in F( x ) f(x)dx 28. 28. a From the CALCULATION point of view : F( x ) = ANTIDERIVATIVE {f(x)} . F(x ANTIDERIVA {f(x) f(x)} b area curv From the GEOMETRIC point of view : F(x) = ∫ f(x)dx = area under the cur ve f(x)dx a of f(x) over [a, b]. . Now let us see integr ation from an ANALYSIS point of view. Let us look at two integ inte examples to see : b ∫ f(x)dx = CHANGE in F( x ) = F( b) − F(a) . F(x F(b) F (a) f(x)dx a Example 1: 2 f(x) = 1 987654321 987654321 987654321 987654321 987654321 987654321 987654321 987654321 987654321 1 0 2 1 2 F(x) = x 1 2 1 0 x 2 1 2 x F(2) F (1) f(x)dx F( ∫ f(x)dx = [ x ] = F( 2) − F(1) = 2 − 1 = 1 . 1 1 Example 2: 1 0 2 f(x f(x) = x 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 098765...
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