1007slides_Part5

1007slides_Part5 - Math 1007, Fall 2006 (These slides...

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Unformatted text preview: Math 1007, Fall 2006 (These slides replace neither the text book nor the lectures.) Part V: Integration & Applications 37. Antiderivatives (p. 110-117) 38. Sigma Notation (118-119) 39. Area Under a Curve and Riemann Sums (120-127) 40. Properties of the Definite Integral (128-129) 41. The Fundamental Theorem of Calculus (130-135) 42. Indefinite Integrals (136-137) 43. Integration by Substitution (138-147) 44. Integration by Substitution for Definite Integrals (148-159) 45. Using symmetry to simplify integrals (160-162) 46. Areas between curves (163-182) 47. Integration by Parts (183-194) 48. Integration by Parts for definite integrals (195-196) 49a. Integration by Partial Fractions: General Set Up (197-199) 49b. Integration by Partial Fractions: Case 1 (200-203) 50. Trigonometric Integrals (204-210) 51. Trigonometric Substitution (211-216) 52. Volume (217-220) 53. Overall Strategy for Integration (221-222) Not covered in Fall 2006: Integration by Partial Fractions (Case 2,3,4) Volume by cylindrical shells 109 37. Antiderivatives Definition: A function F is called an antiderivative of f on an interval I if F ( x ) = f ( x ) for all x in I . Examples: F ( x ) = x 4 4 is an antiderivative of f ( x ) = x 3 since F ( x ) = 4 x 3 4 = x 3 = f ( x ) F ( x ) = ln x is an antiderivative of f ( x ) = 1 x since F ( x ) = 1 x = f ( x ) F ( x ) = tan x is an antiderivative of f ( x ) = sec 2 x since F ( x ) = sec 2 x = f ( x ) F ( x ) = sin x is an antiderivative of f ( x ) = cos x since F ( x ) = cos x = f ( x ) Note that d dx (sin x + 5) = cos x . Hence G ( x ) = sin x + 5 is also an antiderivative of f ( x ) = cos x . In fact, for any constant c , H ( x ) = sin x + c is an antiderivative of f ( x ) = cos x . 110 Theorem: If F is an antiderivative of f on an interval I , then the most general antiderivative of f on I is F ( x ) + c where c is an arbitrary constant. Theorem: Suppose that F and G are both antideriva- tives of f on an intervale I . Then G ( x ) = F ( x ) + c for some constant c . Definition: Z f ( x ) dx is called an indefinite integral of f . Z f ( x ) dx = F ( x ) means F ( x ) = f ( x ) Finding Z f ( x ) dx = F ( x ) means finding the most general antiderivative of f . 111 Examples (over 2 pages) Z 1 · dx = x + c Z k dx = k x + c Z 2 · dx = x 2 + c Z x 4 dx = x 5 5 + c Z x n dx = x n +1 n + 1 + c, n 6 =- 1 Z 1 x dx = ln | x | + c Z e x dx = e x + c Z a x dx = a x ln a + c 112 Z cos x dx = sin x + c Z sin x dx =- cos x + c Z sec 2 x dx = tan x + c Z sec x tan x dx = sec x + c Z csc x cot x dx =- csc x + c Z csc 2 x dx =- cot x + c 113 We can find a particular antiderivative by adding one or more condition that will fix the constant. Example 1: If f ( x ) = 12 x 2- 24 x + 1 and f (1) =- 2, find f ( x )....
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This note was uploaded on 03/22/2010 for the course MATH 1007 taught by Professor Unknown during the Fall '07 term at Carleton.

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1007slides_Part5 - Math 1007, Fall 2006 (These slides...

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