CALCULUSIICH11 - CH APTER 1 FURTH ER TECH NI QUES I N...

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Unformatted text preview: CH APTER 1 FURTH ER TECH NI QUES I N CONSTRUCTI NG ANTI DERI VATI VES CONTENTS 1.1. Review of Definite and I ndefinite I ntegr als 1.2. Tr igonometr ic I ntegr als 1.3. Par tial Fr actions 1.4. I ntegr ation Using Tables and Computer Algebr a Systems 1.5. I mpr oper I ntegr als. 1.1 REVI EW OF DEFI NI TE & I NDEFI NI TE I NTEGRALS 1.1.1. Definite I ntegr als 1.1.2. I ndefinite I ntegr als 1.1.3. The Fundamental Theor em of Calculus 1.1.4. The Substitution Rule 1.1.5. I ntegr ation by Par ts 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 = 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 integr al of f fr om a to b is = = n i i n b a x x f dx x f 1 * ) ( lim ) ( 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 x 1 x 2 . . . x i-1 x i . . . x n-1 b= x n x 1 * x 2 * . . . x i * . . = n i i x x f 1 * ) ( 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 * 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....
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CALCULUSIICH11 - CH APTER 1 FURTH ER TECH NI QUES I N...

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