Integral Proof

Apr 2010
65
2
I have been asked to justify if the following statement is true or false.

'If the integral of f(x) >= 0 on [a,b], then f(x) is >= 0 on [a,b].

Would (8-x^2) over the domain [-4,4] be a counter example to this? thus justifying that the statement is false.

I beleive so as f(x) is <= 0 for some values but the area over that domain is positive. Can anyone tell me if iam on the right path?
 

Prove It

MHF Helper
Aug 2008
12,883
4,999
I have been asked to justify if the following statement is true or false.

'If the integral of f(x) >= 0 on [a,b], then f(x) is >= 0 on [a,b].

Would (8-x^2) over the domain [-4,4] be a counter example to this? thus justifying that the statement is false.

I beleive so as f(x) is <= 0 for some values but the area over that domain is positive. Can anyone tell me if iam on the right path?
\(\displaystyle 8 - x^2\) is NOT \(\displaystyle \geq 0\) for the entire domain \(\displaystyle [-4, 4]\).
 
Apr 2010
65
2
So doesnt that proove that the statement is false?
 

Prove It

MHF Helper
Aug 2008
12,883
4,999
So doesnt that proove that the statement is false?
Definitely not.


I think it should be obvious that if a function is nonnegative for an entire region, then the area between the function and the \(\displaystyle x\) axis can never fall below the \(\displaystyle x\) axis.
 

Defunkt

MHF Hall of Honor
Aug 2009
976
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Israel
I have been asked to justify if the following statement is true or false.

'If the integral of f(x) >= 0 on [a,b], then f(x) is >= 0 on [a,b].

Would (8-x^2) over the domain [-4,4] be a counter example to this? thus justifying that the statement is false.

I beleive so as f(x) is <= 0 for some values but the area over that domain is positive. Can anyone tell me if iam on the right path?
Your counter example is correct, since \(\displaystyle \int_{-4}^4 8-x^2 ~ dx = 8\int_{-4}^4 dx ~ - \int_{-4}^4 x^2 ~ dx = 8\cdot 8 - (\frac{(4)^3}{3} - \frac{(-4)^3}{3}) = 64 - \frac{2}{3}64 = \frac{64}{3} > 0 \)

however \(\displaystyle f(3) = -1 < 0\)


Definitely not.


I think it should be obvious that if a function is nonnegative for an entire region, then the area between the function and the axis can never fall below the axis.
I think you were thinking of the other direction of the question.