If integral equal to zero, then?

In summary, the conversation discusses the validity of the statement \int_0^\infty f(x) dx= 0 implying f(x) = 0 in the domain x\in [0,\infty). The participants consider counterexamples such as odd functions and f(x) = sin(x), concluding that the statement is false and providing examples to support their reasoning.
  • #1
kennethkhoo
10
0

Homework Statement


if this statement is true:
[itex]\int_0^\infty f(x) dx= 0[/itex]
then is this true?
[itex] f(x) = 0[/itex] in domain [itex]x\in [0,\infty)[/itex]


Homework Equations


-NA-


The Attempt at a Solution


Hmm.. I can't come out with a formula that refute that. I would think of an odd function that changes sign in [itex]\infty/2[/itex], except it doesn't exist.

I would think it's false, but i need some example...
 
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  • #2
What if f(x) = sin(x)?
 
  • #3
SteamKing said:
What if f(x) = sin(x)?

Its improper integral is not defined.

ehild
 
  • #4
-deleted: multiple post-
 
  • #5
actually I think it might be er..
[itex] f(x)=sin(x)[/itex] for [itex]x=[0,2\pi][/itex]
[itex] f(x)=0[/itex] for others
 
  • #6
That will do ...

ehild
 

FAQ: If integral equal to zero, then?

What does it mean if an integral equals zero?

An integral is a mathematical concept that represents the area under a curve in a graph. If the integral equals zero, it means that the area under the curve is equal to zero. This could indicate that the function being integrated is symmetric or that the positive and negative areas under the curve cancel each other out.

What are the implications of an integral equaling zero?

The implications of an integral equaling zero depend on the context in which it is used. In some cases, it could mean that the total amount or quantity being measured is equal to zero. In other cases, it could indicate that the function being integrated has certain properties, such as symmetry or periodicity.

How can an integral equaling zero be useful in scientific research?

An integral equaling zero can be useful in scientific research in various ways. It can help determine the symmetry or periodicity of a function, which can provide insights into physical phenomena. It can also be used to solve equations and model real-world situations, such as calculating the displacement of an object over time.

Can an integral ever equal zero if the function being integrated is not equal to zero?

Yes, an integral can equal zero even if the function being integrated is not equal to zero. This can happen when the positive and negative areas under the curve cancel each other out, or when the function has certain properties that result in an integral of zero. However, this does not always mean that the function itself is equal to zero.

How does the concept of an integral equaling zero relate to the concept of derivatives?

An integral equaling zero and the concept of derivatives are closely related. The derivative of a function is the rate of change of that function, while the integral is the reverse process of finding the area under the curve. In some cases, the derivative of a function can be used to determine whether the integral of that function will equal zero or not.

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