Definition of an integral question

[dillon131222]

definition of an integral question...

if f(x) ≥ 0, how can you use the definition of an integral to prove that ∫(a,b)f(x)dx ≥ 0?

This seems like it is an easy question, and seems like one of those things that seems obvious but hard to explain, and the only definition of an integral I've been able to find in my textbook is for a definite integral that I didn't find useful.

 

[Karnage1993]

What was the definition? If you look closely, ∫(a,b)f(x)dx is a definite integral!

 

[dillon131222]

What was the definition? If you look closely, ∫(a,b)f(x)dx is a definite integral!

http://img42.imageshack.us/img42/5464/definitionofintegral.png

was the definition I found in my textbook, which i don't understand all that well, i can evaluate an integral but I am not very good an interpreting definitions like this :/

 

[Vorde]

Are you familiar with the Riemann Sum definition of an integral? Try looking at a Riemann sum when f(x) is always > (or equal to) 0.

Edit: I replied before I saw OP's response, what does f(x) > 0 tell you in the sum?

 

[dillon131222]

Are you familiar with the Riemann Sum definition of an integral? Try looking at a Riemann sum when f(x) is always > (or equal to) 0.

Edit: I replied before I saw OP's response, what does f(x) > 0 tell you in the sum?

no have not heard of a Riemann Sum, but will look into it :)

-.- i don't see a response by OP but alright :P

 

[Karnage1993]

OP = original poster = you!

What Vorde is saying is that you're given that f(x) ≥ 0, and you have the function f in the summation. What can you do from there?

 

[STEMucator]

If you know f(x) ≥ 0 for all x in [a,b], then using Riemann sums it's easy to show that the integral of f(x) over the interval [a,b] must also be ≥ 0.

Start by partitioning your interval [a,b] into n equal sub-intervals. What can you tell me about $f(x_{i}^{*})$ for any choice of $x_{i}^{*}$ in any sub-interval of your choice $[x_{i-1}, x_i]$

 

[dillon131222]

ah k apparently have learned about that, found my notes on riemann sum, just didnt know it by name, from much ealier in the course. going to see what i can come up with with you guys' help so far and my notes :)

 

[dillon131222]

If f(x) ≥ 0 for all x in [a,b], then f(x_i) ≥ 0 and Ʃf(x_i*) ≥ 0.

and if all values for x between [a,b] and Δx = (b-a)/n then Δx ≥ 0 as well.

So, Ʃf(x_i*))Δx ≥ 0. and if all values are ≥ 0, then the limit must be as well?

then ∫(a,b)f(x)dx = lim_n→∞Ʃf(x_i*)Δx ≥ 0

 

[STEMucator]

If f(x) ≥ 0 for all x in [a,b], then f(x_i) ≥ 0 and Ʃf(x_i*) ≥ 0.

and if all values for x between [a,b] and Δx = (b-a)/n then Δx ≥ 0 as well.

So, Ʃf(x_i*))Δx ≥ 0. and if all values are ≥ 0, then the limit must be as well?

then ∫(a,b)f(x)dx = lim_n→∞Ʃf(x_i*)Δx ≥ 0

If f(x) ≥ 0 for all x in [a,b], then f(x_i) ≥ 0 and Ʃf(x_i*) ≥ 0.

Just a small add on. If f(x) ≥ 0 for all x in [a,b], then $f(x_{i}^{*}) ≥ 0$ for any $x_{i}^{*}$ in any sub-interval you choose for all i.

So yes, the sum will behave in the same manner.

Also, Δx is assumed to be positive regardless... it wouldn't make sense for the length of your interval to be negative right?

So the Riemann sum is always greater or equal to zero and even as n→∞, the sum will still be ≥ 0. Though as n→∞, you get the integral over [a,b] which is also ≥ 0 as a result.

 

[iRaid]

http://www.cliffsnotes.com/study_guide/Definite-Integrals.topicArticleId-39909,articleId-39903.html

This page has a pretty good explanation.
Basically you are taking the sum as n goes to infinity (the number of squares) that are infinitesimally small (dx) of some function (usually f(x_i) or f(x_i*))
Which leads to:
$$\lim_{n \to \infty} \sum_{i=1}^{n} f(x_{i*})\Delta x$$

That's my definition :P

 

[dillon131222]

Just a small add on. If f(x) ≥ 0 for all x in [a,b], then $f(x_{i}^{*}) ≥ 0$ for any $x_{i}^{*}$ in any sub-interval you choose for all i.

So yes, the sum will behave in the same manner.

Also, Δx is assumed to be positive regardless... it wouldn't make sense for the length of your interval to be negative right?

So the Riemann sum is always greater or equal to zero and even as n→∞, the sum will still be ≥ 0. Though as n→∞, you get the integral over [a,b] which is also ≥ 0 as a result.

Ah ok, n you good point :P

thanks for the help everyone :)