Squaring the function of integration when given a finite value

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 ^^.

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jbunniii
Homework Helper
Gold Member
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.

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 ^^.
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.

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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.

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jbunniii