Chapter6(1804)

Chapter6(1804) - Ch6/MATH1804/YMC/2009-10 1 Chapter 6...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Ch6/MATH1804/YMC/2009-10 1 Chapter 6. Integration 6.1. The Fundamental Theorem of Calculus The principal theorem of this section is the Fundamental Theorem of Calculus , which is the central theorem of integral calculus. It provides a connection between the operations of differentiation and integration, enabling us to compute integrals using an antiderivative of the integrand function. Definition 6.1 An antiderivative of the function f is a function F such that F ( x ) = f ( x ) Example 6.2 function f ( x ) 1 2 x cos x sin 2 x + 4 x 3 sec 2 x antiderivative F ( x ) x x 2 sin x- 1 2 cos 2 x + x 4 tan x Remark 6.3 Note that if F 1 and F 2 are antiderivatives of f , then F 1 ( x )- F 2 ( x ) = C , where C ∈ R . In other words, if F is an antiderivative of f , then every antiderivative of f has the form F ( x ) + C where C is an arbitrary constant. We call the function F ( x ) + C the indefinite integral of f ( x ) , and we write Z f ( x ) dx = F ( x ) + C where C is a constant Properties of the Indefinite Integration 1. Z k dx = kx + C where k is constant. 2. Z kf ( x ) dx = k Z f ( x ) dx where k is constant. 3. Z ( f ( x ) ± g ( x )) dx = Z f ( x ) dx ± Z g ( x )) dx . Ch6/MATH1804/YMC/2009-10 2 Basic Integration Formulas 1. Z x n dx = x n +1 n + 1 + C for n 6 =- 1 . 2. Z 1 x dx = Z dx x = ln | x | + C . 3. Z e x dx = e x + C Integration Formulas for Trigonometric Functions Z sin xdx =- cos x + C Z cos xdx = sin x + C Z sec 2 xdx = tan x + C Z sec x tan xdx = sec x + C Z csc 2 xdx =- cot x + C Z csc x cot xdx =- csc x + C The definite integral of f ( x ) over the interval [ a,b ] is denoted by Z b a f ( x ) dx. The numbers a and b are called limits of integration ; a is the lower limit and b is the upper limit . Suppose that the graph of f ( x ) over the interval [ a,b ] looks like the following: The geometric interpretation of the definite integral is that Z b a f ( x ) dx = Area of the region bounded by the curve y = f ( x ) , the x-axis, and the lines x = a and x = b Ch6/MATH1804/YMC/2009-10 3 Properties of Definite Integrals 1. Z b a f ( x ) dx =- Z a b f ( x ) dx 2. Z a a f ( x ) dx = 0 3. Z b a k f ( x ) dx = k Z b a f ( x ) dx where k is any number 4. Z b a ( f ( x ) ± g ( x )) dx = Z b a f ( x ) dx ± Z b a g ( x ) dx 5. Z c a f ( x ) dx = Z b a f ( x ) dx + Z c b f ( x ) dx . This property means that the definite integral over an interval can be expressed in terms of definite integrals over subintervals....
View Full Document

This note was uploaded on 11/07/2010 for the course SCIENCE math1804 taught by Professor Prof during the Spring '10 term at HKU.

Page1 / 10

Chapter6(1804) - Ch6/MATH1804/YMC/2009-10 1 Chapter 6...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online