alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

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Unformatted text preview: the “ar ea under the cur v e ” f (x) over the inter val [a,b]. Rectangular method f(x) B (0 , 0 ) f(x) 65432432132 32 44 654321324321321 654321324321321 41 654321324321321 41 654321324321321 41 654321321 1 1 41 654321321321321 14 4 654321 A x 0= a x1 x2 x3 x4 . . . b=xn x Let f(x) be the cur ve between x = a and x = b with f(a) = A and f(b) = B. Divide the interval [a,b] into n sub-intervals ∆ x i and construct rectangles in step-like fashion as shown in the figure. We shall compute F(x) in three steps : n−1 Σ f(x i ).. ∆x i i=0 n →∞ n →∞ i=0 i=0 . . ⇒ Σ f (x i ). δ x i ⇒ L i m i t Σ f (x i ). δx i = discrete summation discrete summation of FINITELY many terms of COUNTABLY many terms 131 b ∫ a continuous summation of UNCOUNTABLY many terms f (x ). dx . APPROXIMATE step discrete summation 1. APPR OXIMATE AREA step : discr ete summation of FINITELY many terms. discr An approximation of the actual area function F(x) is n−1 F(x) = f(x0) (x1 -- a) + f (x1) (x2 -- x1) + ... + f(xn -- 1) (b -- xn -- 1) = Σ f (x i ). ∆x i --. i=0 2. TENDS TO step : discrete summation as n → ∞ of COUNTABLY many terms. step discrete summation discr We let n be very, very large by letting n → ∞ . T he sub-intervals ∆x i = x1 -- a, -x2 -- x1, ..., b -- xn -- 1 will be very, very small. We can denote ∆x i by δx i for i = 0, 1, -2, . . . , n → ∞ . δ x 0 = x 1 -- a, δ x 1 = x 2 -- x 1, -- ..., δ x n -- 1 = b -- x n -- 1 n →∞ So: F(x) = f(x 0) δ x 0 + f (x 1) δ x 1 + . . . + f(x n -- 1) δ x n -- 1 = Σ f (x i ). δ x .i i=0 As n gets larger and larger we can see that the sum is a better and better approximation of the area function F(x). 3. LIMIT step : continuous summation of UNCOUNTABLY many terms. step continuous summation contin Geometrically w e can think of δ x i a s a small continuous l ine segment. c ontinuous As n → ∞ the δ x i gets smaller and smaller. The LIMIT of δ x i a s n → ∞ i s a point or instant. We can denote δ x i by dx . Now we take the LIMIT of the sum as n → ∞ . F(x) = Limit L imit F (x) = Limit L imit Σ f (xi) . dx for i = 0, 1, 2, . . . , n → ∞ . b n →∞ Σ f (xi) . dx = ∫a f (x).. dx i=0 We use the special symbol ∫ w hich is the initial letter of the word summa o r s umma sum, to distinguish the...
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