Integral of f·g ≠ integral f · integral g [True or False]

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In summary, the conversation discusses the statement "integral of f·g ≠ integral f · integral g" and whether it is true or false. One person argues that it is false, while the other argues that it is sometimes true and sometimes false. They also mention the issue of "logical mathematics" and the lack of clear notation for "not necessarily equal to". The conversation concludes by discussing the ambiguity of the statement and whether it is poorly posed.
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Homework Statement
True or False : Integral of f·g ≠ integral f · integral g
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My answer is False! I think must stated "in general," in the beginning of the statement. Cause this could be true if f or g = zero. There may be other cases also.
Is my answer right?
 
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  • #2
If integral of f·g ≠ integral f · integral g is false, then integral of f·g = integral f · integral g.

Thus this make sense?
 
  • #3
Hmmm. Good point! The issue here is that my point is "sometime true sometimes false" & your point "If it not true, then it is always false". huge difference between my point of view and yours! I believe it has something to do with "logical mathematics" if there is something called so.

The issue if you said it is true then I have my counter-example.
 
  • #4
You obviously understand the point, and the answer at this stage is semantics over the implicit qualifier. I think it's a confusing question.
 
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  • #5
There's no clear notation in mathematics for "not necessarily equal to". The best we can do is say something like, for example:

For functions ##f## and ##g## in general ##f \circ g \ne g \circ f##. Even though there are cases where equality holds.
 
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  • #6
in my opinion, if the answer can only be true or false, that implies to me that the statement is universally quantified, so i would argue the correct answer should be false. since the statement is not explicitly quantified however, one could argue that it is poorly posed.
 
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FAQ: Integral of f·g ≠ integral f · integral g [True or False]

1. What is the definition of the integral of f·g?

The integral of f·g is the mathematical operation of finding the area under the curve of the product of two functions, f and g, over a given interval. It is denoted as ∫f·g dx.

2. Is the statement "integral of f·g ≠ integral f · integral g" always true?

No, the statement is not always true. It is only true when the functions f and g are independent of each other, meaning that their values do not affect each other. In this case, the integral of f·g is equal to the product of the integrals of f and g.

3. Can you give an example where the statement is true?

Yes, for example, if f(x) = 2x and g(x) = 3x, then the integral of f·g is ∫2x·3x dx = 6∫x^2 dx = 6(x^3/3) = 2x^3. On the other hand, the integral of f is ∫2x dx = x^2 and the integral of g is ∫3x dx = 3(x^2/2) = 3/2x^2. Therefore, the product of the integrals of f and g is (x^2)(3/2x^2) = 3/2x^4, which is not equal to the integral of f·g.

4. What happens when f and g are not independent?

When f and g are not independent, the statement "integral of f·g ≠ integral f · integral g" is not true. In this case, the integral of f·g is equal to the integral of the product of f and g, plus the integral of the cross-product between f and g. This is known as the Leibniz rule for integration.

5. Can you provide an example where the statement is false?

Yes, for example, if f(x) = x and g(x) = 2x, then the integral of f·g is ∫x·2x dx = 2∫x^2 dx = 2(x^3/3) = 2/3x^3. On the other hand, the integral of f is ∫x dx = x^2/2 and the integral of g is ∫2x dx = x^2. Therefore, the product of the integrals of f and g is (x^2/2)(x^2) = 1/2x^4, which is not equal to the integral of f·g.

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