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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

x f c x i 1 for i 1 2 3 the sign in

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Unformatted text preview: = F(b) − F(a) over [a, b] over [a, b] In both SIGN and MAGNITUDE as we saw in the examples 1 and 2. 2. If f(x) is negative (below the x-axis) then : negative AREA under f(x) = over [a, b] [ F(b) − F(a) ] when the direction of integration direction over [a, b] is positive positive In both SIGN and MAGNITUDE as we saw in the examples 3 and 5. AREA under f(x) = over [a, b] direction − [ F(b) − F(a) ] when the direction of integration over [a, b] is negative negative In both SIGN and MAGNITUDE as we saw in the examples 4 and 6. 155 continuous ov may ind We may use the AREA under continuous f(x) ov er [a, b] to ffind the CHANGE in the INTEGRAL F(x) over [a, b] in the positive direction as follows. positiv 1. We must break up the AREA under f(x) over [a, b] into segments of positive positi areas and negative areas as we did in example 7. negative positive area = area above the x-axis negative area = area below the x-axis Here we implicitly assumed that the direction of integration is positive . direction positive neg tiv positiv 2. We SUM UP all the negative and positive areas to get a net result, say A . Then: ne positi CHANGE in F(x) over [a, b] in the positive direction = F(b) − F(a) = A positive EXERCISES The exercises are to show the relationship between the area under the area cur v e f(x) = F’(x) and the CHANGE IN VALUE of the Integral F(x). Exercise 1: Repeat examples 7 and 8 with f(x) = -- 2x Ex er cise 2: Integrate f(x) = -- 2x over the inter val [-- 2, -- 1] from LEFT to Exer ercise 2 f rom -- 2 to -- 1. What RIGHT. Verify it against the CHANGE in F(x) = -- x is the area under the curve ? Exer ercise Ex er cise 3: Integrate f(x) = -- 2x over the interval [-- 2, -- 1] from RIGHT 2 f rom -- 1 to -- 2. What to LEFT. Verify it against the CHANGE in F(x) = -- x is the area under the curve ? Exercise 4: What is the CHANGE in the value of the function sin (x) over the inter vals [0, π /2], [ π /2, 3π /2], [ π , 2 π ], [0, 2 π ] . Use the fact that the INTEGRAL sin (x) is the ANTIDERIVATIVE of cos (x). Draw the curve of cos (x) over [0, 2π ]. For each interval verify that the CHANGE IN AREA is equal to the CHANGE IN VALUE of sin (x) over the interval going LEFT to RIGHT. Repeat the exercise going RIGHT to LEFT. 156 30. Area under f(x) and Plottin...
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