Integral Question: Deriving Formula for $\int|f(x)|dx$

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In summary, the conversation was about finding the formula or method for deriving the integral of |f(x)| where f(x) is a real, continuous function in the vector space C^\infty. It was mentioned that the area under |f(x)| can be found by breaking up the integral into separate domains and integrating piecewise. There was also some confusion about the notation of f(x) versus x and a humorous comment about reading glasses.
  • #1
Paul
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Hello everyone! Can anyone tell me a formula (or a way to derive) this integral?
[tex]\int|f(x)|dx[/tex]
where [tex] f(x) [/tex] is a real, continuous function of x in the vector space [tex]C^\infty[/tex]. So far, all I've figured out is that odd-order integrations are related to the signum function.
Thanks!
 
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  • #2
Derive or find the area under |f(x)|?

|f(x)| = f(x), if x >=0; -f(x), if x<0

so &Int; |f(x)| dx has 2 solutions: F(x), and -F(X), where F(X) is the antiderivative of f(x).
 
  • #3
PrudensOptimus said:
Derive or find the area under |f(x)|?

|f(x)| = f(x), if x >=0; -f(x), if x<0

so &Int; |f(x)| dx has 2 solutions: F(x), and -F(X), where F(X) is the antiderivative of f(x).

No, |f(x)| = f(x) if f(x) >= 0 and -f(x) if f(x) < 0.

I recommend breaking up the integral into separate domains as I've indicated and integrating piecewise.
 
  • #4
Tide said:
No, |f(x)| = f(x) if f(x) >= 0 and -f(x) if f(x) < 0.

I recommend breaking up the integral into separate domains as I've indicated and integrating piecewise.
Stop repeating my answers :p
 
  • #5
PrudensOptimus said:
Stop repeating my answers :p

LoL! Man, I've just GOTTA get some reading glasses! :bugeye:
 
  • #6
He wasn't repeating your answers. Prudens used x, while Tide correctly used f(x).
 
  • #7
what i really meant was f(x)... but i was thinking about beer.
 

1) What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is used to find the total value or quantity of a function over a certain interval.

2) Why is it important to derive a formula for $\int|f(x)|dx$?

Deriving a formula for $\int|f(x)|dx$ allows us to solve for the area under any curve, even those with negative values. This is important in many real-world applications, such as calculating the total distance traveled by an object with changing velocity, or finding the total amount of profit or loss for a business over time.

3) How do you derive the formula for $\int|f(x)|dx$?

To derive the formula for $\int|f(x)|dx$, we first split the integral into two parts, one for the positive values of the function and one for the negative values. Then, we use the definition of absolute value to convert the negative values into positive values. Finally, we integrate both parts separately and combine them back together to get the final formula.

4) Can the formula for $\int|f(x)|dx$ be applied to all functions?

Yes, the formula for $\int|f(x)|dx$ can be applied to all functions, as long as the function is continuous and defined over a finite interval. It is a general formula that can be used for any function, regardless of its shape or type.

5) Why is it important to use absolute value when deriving the formula for $\int|f(x)|dx$?

Using absolute value ensures that we are calculating the entire area under the curve, including any negative values. Without it, the integral would only calculate the positive area, leading to an incorrect result. Additionally, using absolute value allows us to simplify the integral and make it easier to solve.

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