# Absolute Value Integral Proof

• katia11
In summary, the given conversation discusses a proof involving the continuity of a function. The hint provided suggests using the properties of continuous functions, such as the fact that if f is continuous, then |f| is also continuous. The proof should be approached using Proof by Contradiction, and one should consider the value of |f(x)| when x is close to c.

## Homework Statement

Prove that, if f is continuous on [a,b] and

ab= l f(x) l dx = zero

then f(x) = 0 for all x in [a ,b].

## Homework Equations

Hint- from book-
Section 2.4 Exercise 50
Let f and g be continuous at c. Prove that if:
(a) f(c) > o, then there exists delta > o such that f(x) > 0 for all x E (c- delta, c + delta)
(b) f(c) < o, then there exists delta > o such that f(x) < 0 for all x E (c- delta, c + delta)
(c) f(c) < g(c) , then there exists delta > o such that f(x) < g(x) for all x E (c- delta, c + delta)

## The Attempt at a Solution

I understand what we are trying to prove, I can visualize it, but I have NO idea what the "hint" has to do anything. I'm really not a "math person" I discovered. That is a terrible realization. I'm not just looking for the answer though, I just have no idea where to start and proofs scare me.

katia11 said:

## Homework Statement

Prove that, if f is continuous on [a,b] and

ab= l f(x) l dx = zero

then f(x) = 0 for all x in [a ,b].

## Homework Equations

Hint- from book-
Section 2.4 Exercise 50
Let f and g be continuous at c. Prove that if:
(a) f(c) > o, then there exists delta > o such that f(x) > 0 for all x E (c- delta, c + delta)
(b) f(c) < o, then there exists delta > o such that f(x) < 0 for all x E (c- delta, c + delta)
(c) f(c) < g(c) , then there exists delta > o such that f(x) < g(x) for all x E (c- delta, c + delta)

## The Attempt at a Solution

I understand what we are trying to prove, I can visualize it, but I have NO idea what the "hint" has to do anything. I'm really not a "math person" I discovered. That is a terrible realization. I'm not just looking for the answer though, I just have no idea where to start and proofs scare me.

The hint tells you that:
• If f(c) < 0. Then f(x) < 0, for x close enough to c.
• If f(c) > 0. Then f(x) > 0, for x close enough to c.

Can you prove these two hints? These two are very useful when dealing with continuous functions.

Well, when tackling some problem with so little information like this, one should think right about: Proof by Contradiction. There's a small property (theorem) that you should know:
$$\mbox{If } f \mbox{ is continuous, then } |f| \mbox{ is also continuous.}$$​

The theorem above should be easy to prove using delta-epsilon method. Let's see if you can get it.

Ok, so back to the main problem. As I said earlier, you should use Proof by Contradiction. I'll give you a little push.

Assume that: $$\exists c \in [a; b] : |f(c)| \neq 0$$, so, what can you say about the value of |f(x)|, when x is close enough to c?

## What is the concept of absolute value integral?

The absolute value integral is a mathematical concept that involves finding the area under a curve on a given interval, taking into account both positive and negative values. It is represented by the vertical distance between the curve and the x-axis, and is always considered as positive.

## How is the absolute value integral represented mathematically?

The absolute value integral is represented by the notation ∫|f(x)| dx, where f(x) is the function whose area is being calculated and dx represents the infinitesimal change in x. The absolute value bars ensure that the integral always yields a positive value.

## What is the importance of the absolute value integral?

The absolute value integral is important because it allows us to find the total area under a curve, even when the function has both positive and negative values. It is often used in real-world applications, such as calculating displacement, velocity, and acceleration in physics.

## How is the absolute value integral calculated?

The absolute value integral is calculated by breaking down the given interval into smaller parts, approximating the curve with rectangles, and then taking the sum of the areas of these rectangles. As the width of the rectangles decreases, the accuracy of the calculation increases.

## What are some common techniques used to solve absolute value integral problems?

Some common techniques used to solve absolute value integral problems include using the Fundamental Theorem of Calculus, substitution, and integration by parts. It is also important to understand the properties of integrals, such as linearity and the mean value theorem, to effectively solve these types of problems.