# Squaring the function of integration when given a finite value

1. May 9, 2009

### Iwonder9000

I have a problem with this one question that is driving me crazy. It seems so simple. I have done definite integrals, 'U' substitutions, and integration by parts, etc, etc... Anyway, here's the question, maybe you guys can help me out. This isn't a homework question or anything, it just has been on my nerves.

Alright so I am given: int(f(x)) from 1 to 10 = 4. And they want me to find out what the int(f(x)^2) from 1 to 10 is... Now I've tried doing a host of different things but I been unable to draw a conclusion as to how this can be solved. If anyone can help me with this simple problem it would be much appreciated.

Note: This isn't int(f(x))^2 = 4, which is 16.

Picture:

http://img16.imageshack.us/img16/580/int101.jpg [Broken]

Thanks in advance. And, I also apologize if I have gone against any rules or anything of that nature. After all this is my first post ^^.

Last edited by a moderator: May 4, 2017
2. May 9, 2009

### jbunniii

You don't have enough information to answer the question.

Consider the following two functions:

$$f_1(x) = 1$$ for $$1 \leq x \leq 4$$ and 0 elsewhere

and

$$f_2(x) = 2$$ for $$1 \leq x \leq 2$$ and 0 elsewhere

Then

$$\int_1^{10} f_1(x) dx = \int_1^{10} f_2(x) dx = 4$$

but

$$\int_1^{10} (f_1(x))^2 dx = 4$$

whereas

$$\int_1^{10} (f_2(x))^2 dx = 8$$

You can at least find a lower bound for your integral, because $$f(x)^2 \geq 0$$.

But there is no upper bound: consider

$$f_n(x) = 4n$$ for $$0 \leq x \leq 1/n$$ and 0 elsewhere

Then

$$\int_1^{10} f_n(x) dx = 4$$

but

$$\int_1^{10} (f_n(x))^2 dx = 16n$$

so the square can be made as big as you like.

Last edited: May 9, 2009
3. May 9, 2009

### Iwonder9000

Thank you for your reply. However, there should be an answer to this question since it was on my test a while back. The semester is over, but I never got around to asking my professor what the answer could be. I have notes in front of me, and if it's of any use the answer is definitely not 16 or 8. Hmmm... personally, the more I think about it maybe this wasn't a fair question to put on an exam. I may have to email my professor or something on this one. Either way, if I find out the answer I definitely post it.

Last edited: May 9, 2009
4. May 9, 2009

### jbunniii

The question as you stated it doesn't have a unique answer, but maybe on the test there was an additional constraint that would have made it so. If you don't have a copy of the exam anymore, I'm sure the professor wouldn't mind if you e-mailed him and asked him about the question, especially if it used some trick that he is proud of.

5. May 9, 2009

### Iwonder9000

Hmmm, you know what, I just over looked the problem again, and in small print it says "determine the following values, if possible". That could very well mean that you are right, there is no answer, simply saying this problem is unsolvable given the information could have done the trick. I will still email my professor to be sure, but now it almost seems as if this problem is definitely unsolvable given what little information I have to go on. Thank you again for your input, it has put me in the right direction.

6. May 12, 2009

### Iwonder9000

Update(Sorry for the double post):

I emailed my professor and you are correct, it is unsolvable. Thank you again.