alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

In the diagram below let the intervals t1 t2 and t2

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Unformatted text preview: 4321 0987654321 0987654321 2 4 1 2 2 F(x) = x 2 2 3 22 2 2 1 x 0 12 2 1 2 3 4 x F(2) F (1) f(x)dx F( ∫ f(x)dx = [ 1/2 x 2 ] = F( 2) − F(1) = 1/2 (2 ) − 1/2 (1 ) = 3/2 . 1 1 2 138 2 In general: b F(x CHANGE in F( x ) = ∫ f(x)dx = f(x)dx What is ∫ a b a F(b) F (a) F( b) − F(a) F( F(a) F (b) f(x)dx ? It must be F( a) − F(b) . f(x)dx F( Apply this to the two examples we just saw. 1 1 F(1) F (2) f(x)dx F( ∫ f(x)dx = [ x ] = F( 1) − F(2) = 1 − 2 = − 1 . 1 2 2 1 2 2 F(1) F (2) f(x)dx F( ∫ f(x)dx = [ 1/2 x 2 ] = F( 1) − F(2) = 1/2 (1 ) − 1/2 (2 ) = − 3/2 . 2 2 area curv Implying that the ar ea under the cur ve of f(x) is NEGATIVE. We know from ar ea. middle school Geometry that there is no such thing as a neg ati ve area So how neg tiv area do we explain this result ? integ In integr ation the DIRECTION OF THE INTEGRATION has a role to play as we inte shall see in the next chapter. Moreover, this aspect gives the correct physical interpretation. Calculation: Derivative f(x) Geometric: Integrand f(x) Analysis: Anti-Derivative d F(x) dx DIFFERENTIATION ∫ f(x)dx f(x)dx INTEGRATION Instantaneous Rate of Change 139 F(x) Integral F(x) Change 29. Direction Integ Direction of Integr ation and CHANGE in F(x) and Let F(x) = ∫ f(x)dx When we inte g r a te the function f(x) over the inter val [a,b] we get a integ value or NUMBER. From the Calculus point of view this NUMBER represents a CHANGE in the value of the integral F(x) from a to b. b b ∫ a f(x)dx = [F(x)] a = F (b) -- F(a) The NUMBER may be positive (> O), meaning an INCREASE in the value of F(x) from a to b. The NUMBER may be negative (< O), meaning a DECREASE in the value of F(x) from a to b. The NUMBER may be zero, meaning there is NO CHANGE in the value of F(x) from a to b . Geometrically speaking there is no such thing as a negative area or negative length. “area curv We may use the “area under the cur ve” f(x) over the interval [ a, b] calculation “ar to find the CHANGE in F(x). We follow the simple rules in the table below to determine the SIGN of the “area under the cur ve” f(x) in our calculation. “area curv “ar SIGN of “area curv “area under the cur ve...
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