Let

f(x)=

{ 1 if -1 < or equal to x<0;

{-1 if 0 < or equal to x < or equal to 1.

Prove that f(x) is integrable on -1 < or equal to x < or equal to 1.

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- Dec 19th 2006, 12:01 AMSwamifezHelp with Integrable Proof
Let

f(x)=

{ 1 if -1 < or equal to x<0;

{-1 if 0 < or equal to x < or equal to 1.

Prove that f(x) is integrable on -1 < or equal to x < or equal to 1. - Dec 19th 2006, 02:07 AMCaptainBlack
I fail to see how this can be a problem, what do you know about integration?

(what type of integral are we talking about here).

If this is a Riemann integral it is obvious that the integral of is

defined over from a elementary consideration of the Riemann

sums of for partitions of this interval, and that the integral is:

RonL - Dec 19th 2006, 06:50 AMThePerfectHacker
- Dec 21st 2006, 04:04 AMSwamifez
Sorry I am posting so much, but these problems are going to be similar to by final. Can someone help me clarify this problem in to one big proof. I also need to pick 5 paritions of this interval and prove it within. Thank you

- Dec 21st 2006, 09:33 AMThePerfectHacker
Okay, for suffiently large .

You want to find,

I will use right endpoints,

But,

Thus,

Note,

(greatest integer).

Thus, the function is 1 for these values.

And it is -1 for after these values.

If is even we have,

Thus, as the limit of the Riemann sum is zero.

If is odd we have,

Thus, as the limit of the Riemann sum is zero.

Thus, the limit is always zero no matter what.