show that if "f" is integrable then

f(x) < or equal to 0 on [a,b]

http://upload.wikimedia.org/math/1/8...fb77870925.png is greater than or = to zero

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- Mar 7th 2010, 12:16 PM10tomlinsonbintegrals of nonpositive functions
show that if "f" is integrable then

f(x) < or equal to 0 on [a,b]

http://upload.wikimedia.org/math/1/8...fb77870925.png is greater than or = to zero - Mar 7th 2010, 12:29 PMPlato
- Mar 7th 2010, 12:36 PM10tomlinsonb
sorry that is what i meant to write. can you show me why that is true?

- Mar 7th 2010, 12:44 PMPlato
- Mar 7th 2010, 01:03 PM10tomlinsonb
it is a problem in definite integrals

- Mar 7th 2010, 01:16 PMPlato
- Mar 7th 2010, 01:29 PM10tomlinsonb
can you show me how you would prove this using upper sums

- Mar 7th 2010, 02:08 PM10tomlinsonb
please help me prove this because i am so confused and i need it for a homework problem

- Mar 7th 2010, 02:30 PMHallsofIvy
The reason Plato asked if you knew about Riemann sums is that how you prove such a basic property of integrals depends upon how you

**define**integrals in the first place! And "Riemann sums" is a common method of defining integrals. You mention of "upper sums" indicates that's what you have learned, although "upper sums" is only a very small part of that.

Given that f(x) is integrable on [a, b], then for any partition of [a, b] (dividing it into smaller subintervals), choosing any in the interval, with width , the Riemann sum is and, as we take "finer and finer" partitions, meaning that the largest goes to 0 as we take more and more intervals in the partition, and that sum goes to some fixed value, I, whic**is**the integral, .

Now, if for all x in [a, b], in particular, for every partition and for every in a partition. That, in turn, means that for every partition and every i in a partition, and so for every partition. The integral, the limit of negative numbers can't be positive: . - Mar 7th 2010, 02:57 PM10tomlinsonb
my teacher also told that

(b-a) is max of f(x) therefore,

http://www.mathhelpforum.com/math-he...8b16c3ed-1.gif ≤ (b-a) ≤ 0

can you expain why this is - Mar 8th 2010, 04:41 AMCaptainBlack