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CALCULUSIICH11 - C H APT ER 1 F U RT H ER T ECH N I QU ES I...

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CHAPTER 1 FURTHER TECHNI QUES I N CONSTRUCTI NG ANTI DERI VATI VES
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CONTENTS 1.1. Review of Definite and Indefinite I ntegrals 1.2. Trigonometric Integrals 1.3. Partial Fractions 1.4. I ntegration Using Tables and Computer Algebra Systems 1.5. I mproper I ntegrals.
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1.1 REVIEW OF DEFI NI TE & I NDEFI NI TE INTEGRALS 1.1.1. Definite Integrals 1.1.2. I ndefinite Integrals 1.1.3. The Fundamental Theorem of Calculus 1.1.4. The Substitution Rule 1.1.5. I ntegration by Parts
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1.1.1. Definite Integrals 1.1.1. Definite Integrals Recall Recall that that Definition. if f is a continuous function on the interval [ a , b ], then we divide [ a , b ] into n subintervals of equal width x= (b-a)/n with end points x 0 = a , x 1 , x 2 , …, x n = b . Let x 1 * , x 2 * , …, x n * be sample points selected in these subintervals. Then the definite integral of f from a to b is = = n i i n b a x x f dx x f 1 * ) ( lim ) (
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Note 2 . The sum is called the Riemann sum It can be interpreted as the sum of the area of the green rectangles. a= x 0 x 1 x 2 . . . x i-1 x i . . . x n-1 b= x n x 1 * x 2 * . . . x i * . . . x = n i i x x f 1 * ) (
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Hence in the case where f is positive, by passing to the limit, we see that the area of the region under the curve y= f(x) is given by = = = b a n i i n dx x f x x f A ) ( ) ( lim 1 *
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In case f takes both + and – signs, the Riemann sum is the sum of the areas of the green rectangles minus the sum of the areas of the gold rectangles. a b
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Hence the definite integral may be interpreted as the "signed area" of the region limited by the curve y= f(x), the x - axis and the lines x = a and x = b, where the region above the x -axis (green) is counted with + sign and the region below the x -axis (gold) is counted with – sign a b - - + +
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