alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

S i n t 1 g t 2 5 2 5 t 2 t case 2 5 meters above

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Unformatted text preview: : positive direction b F (a) ∫ f(x)dx = F(b) -- F(a) a b F(b F(a F( b) = ∫ f(x)dx + F( a) a a If F( b) is known, then from b going RIGHT to LEFT in the negative direction to a : F(b negative direction a F(b F(a F( ∫b f(x)dx = F( a ) -- F( b ) a F(a) = ∫b f(x)dx + F(b) Using the relations above we may compute F C ( x) . We must choose a small inter val x . Then {f(x). ∆ x} is an element of ar ea under the area . ar cur v e o f f(x) over the inter val ∆ x . The smaller the ∆ x the more accurate the approximation of the ar ea under the cur v e o f f(x) over the interval area ar ∆ x . Another reason why we have to choose ∆ x very small is that we do not skip or miss the EXTREMUM points (maximum and minimum) in plotting F C ( x). These are the points where f(x) CHANGES SIGN. next FC (x) = CHANGE in FC (x) over next ∆x + current value of FC (x) CHANGE in FC (x) over next ∆x = area under the curve of f(x) over next ∆x area curv ar 159 Going RIGHT (in the postive direction ) from x 0 with i = 1 : p ostive w ith F C (x i ) = (x ∫ x x + ∆x i −1 f(x) dx + F C (x i − 1 ) f (x) (x i−1 +∆ (x (x F C (x i ) ≅ f (x i ) .( +∆ x ) + F C (x i − 1 ) for i = 1, 2, 3, . . . The + SIGN in ( + ∆ x) is due to the postive direction i n the integration. p ostive Going LEFT (in the negative direction ) from x 0 with i = − 1: n egative w ith F C (x i ) = (x ∫ x x − ∆x i +1 f(x) dx + F C (x i +1 ) f (x) (x i +1 (x (x F C (x i ) ≅ f(x i ) .( − ∆ x ) + F C (x i + 1 ) for i = −1, −2, −3, . . . The − SIGN in ( − ∆x) is due to the negative direction in the integration. negative This is the r ectangular method o f NUMERICAL INTEGRATION. In the next chapter we shall see how to compute and interpret the meaning of the CONSTANT OF INTEGRATION from the nature of the problem. 160 EXERCISES The exercises are to show how to plot FC ( x) when only f(x), x 0 a nd FC ( x 0) are known and the constant of integration C is not known. Also, we do not need to know the expression of F(x). We make use of t...
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