alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

The sign reflects an increase in fx over 1 2 in the

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Unformatted text preview: i x In the LIMIT as n → ∞ e ach trapezoidal infinitesimal element o f area i nfinitesimal -- ) . δ x b ecomes an instantaneous element o f area f(x)dx . f( x i i nstantaneous i F(x) = ∫ f (x)dx = ∫ 2 x dx = x 2 T he AREA under f(x) over the inter v al [1, 2] going LEFT to RIGHT in the positive directio n i s : p ositive direction 2 2 2 1 1 1 ∫ f (x)dx = ∫ 2 x dx = [x 2 ] 2 [x 2 ] 1 = ( 2) 2 -- (1) 2 = + 3 -- The AREA is 3 in magnitude. The + SIGN is due to (positive f(x)) . (positive direction ) . F(2) -- F(1) = (2) 2 -- (1) 2 = + 3 -The CHANGE in F(x) is 3 in magnitude. The + SIGN reflects an INCREASE in F(x) over [1, 2] in the positive direction. positive 147 f(x) = 2x 3 2 1 δx -- 2 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 -- 3 -- 1 0 -- 1 -- 2 1 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 4 δx F(x) = x 2 2 3 x Note: f(x) δx is an infinitesimal element of area. f(x)dx is an i nstantaneous element o f area. If we try to depict it, it will appear as a line. -- 3 We know from mensuration that the area of a trapezoid is = 1/2 (sum of the parallel sides) . (perpendicular distance). This is the trapezoidal method of NUMERICAL INTEGRATION. We do not need to know -- i . However, we must be x careful to ensure that the parallel sides f ( x i -- 1 ) and f ( x i ) of the trapezium are of the same SIGN. So we must watch out for the points where f(x) CHANGES SIGN. In this example f(x) CHANGES SIGN at x = 0. 148 Example 2: Integr ate contin uous f (x) over the inter val [1, 2] going continuous c ontin RIGHT to LEFT in the negative directio n . n egative direction The AREA under f(x) over the inter val [1, 2] going RIGHT to LEFT in the negative direction i s : n egative 1 ∫ f (x)dx = ∫ 2 2 [x ] 1 2 1 2 2 x dx = [x 2 ] 2 = ( 1) -- (2) -- 2 1 2 = -- 3 -- The AREA is 3 in magnitude. The -- SIGN is due to (positive f(x)) . (negative directio...
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