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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# We sum up all the negative and positive areas to get

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Unformatted text preview: s due to (negative f(x)) . (positive direction ) . 150 4 Note: f(x) δx is an infinitesimal element of area. f(x) = -- 2x δ x -- 1 2 1 δx 0 -- 1 -- 2 -- 3 F(x) = -- x 2 f(x)dx is an i nstantaneous element o f area. If we try to depict it, it will appear as a line. -- 4 151 151 1 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 -- 2 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 321 -- 3 3 2 3 x F(2) -- F(1) = -- (2) 2 -- (-- (1) 2 ) = -- 3 The CHANGE in F(x) is 3 in magnitude. The -- SIGN reflects a DECREASE in F(x) over [1, 2] in the positive direction . positive continuous Example 6: Integrate contin uous f (x) over the inter val [1, 2] going c ontin RIGHT to LEFT in the negative direction . negative T he AREA under f(x) over the inter val [1, 2] going RIGHT to LEFT in the negative direction i s : negative ∫ 1 2 f (x)dx = ∫ 1 [-- x 2 ] 2 1 2 -- 2x dx = [-- x 2 ] - 1 2 = -- (1) 2 -- ( -- (2) 2 ) = -- 1 + 4 = + 3 The AREA is 3 in magnitude. The + SIGN is due to (negative f(x)) . (negative direction ) . F (1) -- F(2) = -- (1)2 -- (-- (2) 2 ) = + 3 The CHANGE in F(x) is 3 in magnitude. The + SIGN reflects an INCREASE in F(x) over [1, 2] in the negative direction. negative a b b a ∫ f (x)dx = -- ∫ -- f (x)dx RIGHT to LEFT = -- LEFT to RIGHT -over [a,b] over [a,b] 152 In all the previous examples f(x) did not CHANGE SIGN over the interval of integration. Now let us see two examples where f(x) CHANGES SIGN over the interval of integration. continuous --1, c ontin -Example 7: Integrate contin uous f (x) = 2x over the inter val [-- 2] going LEFT to RIGHT in the positive direction . See diagram page 148. positive The AREA under f(x) = 2x over the inter val [-- 2] going LEFT to RIGHT --1, -in the positive direction i s : positive 2 0 2 --1 -- --1 -- 0 ∫ 2x dx = ∫ 2x dx + ∫ 2 x dx We must brea...
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