   Chapter 4, Problem 16PS

Chapter
Section
Textbook Problem

Proof Prove that if f is a continuous function on a closed interval [a, b], then | ∫ a b f ( x ) d x | ≤ ∫ a b | f ( x ) | d x .

To determine

To prove: The given statement | abf(x)dx |ab| f(x) |dx

Explanation

Given:

For any continuous function f(x) on [ a,b ]

| abf(x)dx |ab| f(x) |dx

Proof:

Consider the absolute value property of any function f(x)

| f(x) |f(x)| f(x) |

Consider the properties that for two integral functions f(x),g(x) with f(x)g(x)

abf(x)dxabg(x)dx

Consider f(x)| f(x) | therefore,

abf

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