alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

The sign is due to negative fx negative direction 2

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Unformatted text preview: n ) . F(1) -- F(2) = (1) 2 -- (2) 2 = -- 3 --The CHANGE in F(x) is 3 in magnitude. The -- SIGN reflects a DECREASE in F(x) over [1, 2] in the negative direction . negative 3: continuous Example 3: Integrate continuous f (x) over the inter val [--2, --1] going contin LEFT to RIGHT in the positive direction . positive The AREA under f(x) over the inter val [--2, --1] going LEFT to RIGHT in the positive direction i s : ∫ --1 ---2 [x 2 ] f (x)dx = ∫ --1 ----2 --1 ---2 2x dx = [x 2 ] --1 ---- 2 = (-- 1) 2 -- (-- 2) 2 = -- 3 -- - The AREA is 3 in magnitude. The -- SIGN is due to (negative f(x)) . (positive direction ) . F (-- 1) -- F(-- 2) = (-- 1) 2 -- (-- 2) 2 = -- 3 -- The CHANGE in F(x) is 3 in magnitude. The -- SIGN reflects a DECREASE in F(x) over [--2, --1] in the positive direction . positive 149 4: continuous --2, Example 4: Integrate continuous f (x) over the inter val [-- -- going contin -- --1] RIGHT to LEFT in the negative direction . negative The AREA under f(x) over the interval [-- -- going RIGHT to LEFT in the in --2, -- --1] the negative direction is : negative -- 2 ∫ --1 -- 2 f (x)dx = ∫ -- 2 [x 2 ] --1 --1 -- 2 2x dx = [x 2 ] --1 = ( -- 2) 2 -- (-- 1) 2 = +3 -- - The AREA is 3 in magnitude. The + SIGN is due to (negative f(x)) . (negative 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 [-- -- in the negative direction. --2, negative -- --1] Let us now consider F(x) = -- x 2 . Then f(x) = F ’ (x) = -- 2 x and verify our calculations with the graph. continuous c ontin Example 5: Integrate contin uous f (x) over the inter val [1, 2] going LEFT to RIGHT in the positive direction . p ositive T he AREA under f(x) over the inter val [1, 2] going LEFT to RIGHT in the positive direction i s : p ositive 2 2 1 1 ∫ f (x)dx = ∫ [-- x 2 ] - 2 1 -- 2x dx = [-- x 2 ] -- 2 1 = -- (2) 2 -- ( -- (1) 2 ) = -- 4 -- (-- 1) = -- 3 - The AREA is 3 in magnitude. The -- SIGN i...
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