This preview shows pages 1–7. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 1007, Fall 2006 (These slides replace neither the text book nor the lectures.) Part V: Integration & Applications 37. Antiderivatives (p. 110117) 38. Sigma Notation (118119) 39. Area Under a Curve and Riemann Sums (120127) 40. Properties of the Definite Integral (128129) 41. The Fundamental Theorem of Calculus (130135) 42. Indefinite Integrals (136137) 43. Integration by Substitution (138147) 44. Integration by Substitution for Definite Integrals (148159) 45. Using symmetry to simplify integrals (160162) 46. Areas between curves (163182) 47. Integration by Parts (183194) 48. Integration by Parts for definite integrals (195196) 49a. Integration by Partial Fractions: General Set Up (197199) 49b. Integration by Partial Fractions: Case 1 (200203) 50. Trigonometric Integrals (204210) 51. Trigonometric Substitution (211216) 52. Volume (217220) 53. Overall Strategy for Integration (221222) 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 )....
View
Full
Document
This note was uploaded on 03/22/2010 for the course MATH 1007 taught by Professor Unknown during the Fall '07 term at Carleton.
 Fall '07
 Unknown
 Calculus, Antiderivatives, Derivative, Riemann Sums

Click to edit the document details