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Studying Mathematics Thread
#84
RE: Studying Mathematics Thread
(April 20, 2018 at 10:39 pm)Grandizer Wrote:
(April 20, 2018 at 11:10 am)Grandizer Wrote: Ok, now I'm confused about the following:

The improper integral of 1/x from -e to e is said to diverge yet can be assigned a CPV value of 0.

Intuitively, the 0 answer makes sense (since symmetrically opposite areas, even infinite, should completely cancel each other out), but it's considered to be a problematic answer. Can anyone ELI5 why 0 is not always the right answer here? What does definite integral really mean then?

After some reading, I think I have my answer:

Intuitively speaking, there is nothing wrong with the thinking that, even if infinite, symmetrical and opposite arithmetic areas cancel each other out. And this accords well with the CPV. However, mathematically speaking, the way definite integrals are defined, symmetry isn't assumed by the definition. So when dealing with improper integrals (like the one above), one can arbitrarily choose whatever values they want close to 0 (one to the left of 0 and the other to its right), and so the values need not be opposites of each other. Since that's the case, when calculating the limit, it could be anything depending on the arbitrary values chosen as the bounds closest to 0 (e.g., 0 in the case of a and -a, ln(2) in the case of a and -2a, ln(3) in the case of a and -3a).

A roughly similar situation happens with integrals from -infinity to infinity.

So, what is the integral from -infinity to infinity of x dx?

If you do it symmetrically, from -b to b, and take the limit, the answer is 0.

If you do it from -b to b+1/b, and take the limit as b goes to infinity, you will get 2.

If you do it from -b to b+1, you will get infinity.

Play around with the function 2x/(x^2 +1). This function goes to 0 at infinity, but the doubly infinite integral is still undefined since the positive and negative are both infinite.

The symmetric integral is 0.

The integral going from -b to b^2 gives 2.

The integral from -b to b^3 gives 3.

Such is the issue with divergent integrals. They are like conditionally convergent series: you can get any value out of them if you play enough.
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Messages In This Thread
Studying Mathematics Thread - by Grandizer - February 28, 2018 at 11:22 am
RE: Studying Mathematics Thread - by Silver - February 28, 2018 at 11:31 am
RE: Studying Mathematics Thread - by Tiberius - February 28, 2018 at 11:36 am
RE: Studying Mathematics Thread - by Whateverist - February 28, 2018 at 3:58 pm
RE: Studying Mathematics Thread - by polymath257 - February 28, 2018 at 11:45 am
RE: Studying Mathematics Thread - by robvalue - February 28, 2018 at 1:21 pm
RE: Studying Mathematics Thread - by polymath257 - February 28, 2018 at 2:08 pm
RE: Studying Mathematics Thread - by Edwardo Piet - February 28, 2018 at 2:10 pm
RE: Studying Mathematics Thread - by polymath257 - February 28, 2018 at 2:35 pm
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - February 28, 2018 at 8:55 pm
RE: Studying Mathematics Thread - by Grandizer - February 28, 2018 at 9:00 pm
RE: Studying Mathematics Thread - by Edwardo Piet - February 28, 2018 at 9:02 pm
RE: Studying Mathematics Thread - by Fireball - February 28, 2018 at 3:53 pm
RE: Studying Mathematics Thread - by Alexmahone - February 28, 2018 at 3:54 pm
RE: Studying Mathematics Thread - by LadyForCamus - February 28, 2018 at 10:22 pm
RE: Studying Mathematics Thread - by Grandizer - February 28, 2018 at 10:59 pm
RE: Studying Mathematics Thread - by polymath257 - March 3, 2018 at 7:12 pm
RE: Studying Mathematics Thread - by Grandizer - March 3, 2018 at 7:52 pm
RE: Studying Mathematics Thread - by polymath257 - March 3, 2018 at 9:00 pm
RE: Studying Mathematics Thread - by robvalue - March 4, 2018 at 7:12 am
RE: Studying Mathematics Thread - by Grandizer - March 6, 2018 at 7:58 am
RE: Studying Mathematics Thread - by polymath257 - March 6, 2018 at 9:21 am
RE: Studying Mathematics Thread - by robvalue - March 7, 2018 at 1:51 am
RE: Studying Mathematics Thread - by polymath257 - March 7, 2018 at 8:19 am
RE: Studying Mathematics Thread - by Grandizer - March 7, 2018 at 8:41 am
RE: Studying Mathematics Thread - by robvalue - March 7, 2018 at 8:48 am
RE: Studying Mathematics Thread - by polymath257 - March 7, 2018 at 8:49 am
RE: Studying Mathematics Thread - by robvalue - March 7, 2018 at 8:30 am
RE: Studying Mathematics Thread - by polymath257 - March 7, 2018 at 8:42 am
RE: Studying Mathematics Thread - by robvalue - March 7, 2018 at 8:53 am
RE: Studying Mathematics Thread - by polymath257 - March 7, 2018 at 9:01 am
RE: Studying Mathematics Thread - by robvalue - March 7, 2018 at 10:19 am
RE: Studying Mathematics Thread - by polymath257 - March 7, 2018 at 10:43 am
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - March 8, 2018 at 1:54 am
RE: Studying Mathematics Thread - by robvalue - March 8, 2018 at 2:46 am
RE: Studying Mathematics Thread - by Grandizer - March 8, 2018 at 3:39 am
RE: Studying Mathematics Thread - by robvalue - March 8, 2018 at 3:49 am
RE: Studying Mathematics Thread - by Grandizer - March 8, 2018 at 3:53 am
RE: Studying Mathematics Thread - by polymath257 - March 8, 2018 at 10:47 am
RE: Studying Mathematics Thread - by Edwardo Piet - April 20, 2018 at 10:22 am
RE: Studying Mathematics Thread - by Grandizer - April 20, 2018 at 11:10 am
RE: Studying Mathematics Thread - by Grandizer - April 20, 2018 at 10:39 pm
RE: Studying Mathematics Thread - by polymath257 - April 21, 2018 at 10:06 am
RE: Studying Mathematics Thread - by robvalue - March 8, 2018 at 4:00 am
RE: Studying Mathematics Thread - by Grandizer - March 8, 2018 at 4:11 am
RE: Studying Mathematics Thread - by Grandizer - March 8, 2018 at 8:43 am
RE: Studying Mathematics Thread - by robvalue - March 8, 2018 at 9:48 am
RE: Studying Mathematics Thread - by Grandizer - March 15, 2018 at 1:49 am
RE: Studying Mathematics Thread - by polymath257 - March 18, 2018 at 10:43 pm
RE: Studying Mathematics Thread - by Grandizer - March 19, 2018 at 9:11 am
RE: Studying Mathematics Thread - by polymath257 - March 19, 2018 at 9:32 pm
RE: Studying Mathematics Thread - by polymath257 - March 22, 2018 at 9:34 am
RE: Studying Mathematics Thread - by Grandizer - March 22, 2018 at 10:06 am
RE: Studying Mathematics Thread - by Grandizer - March 22, 2018 at 9:58 pm
RE: Studying Mathematics Thread - by polymath257 - March 23, 2018 at 7:44 am
RE: Studying Mathematics Thread - by Grandizer - March 24, 2018 at 8:45 am
RE: Studying Mathematics Thread - by polymath257 - March 24, 2018 at 9:52 am
RE: Studying Mathematics Thread - by Grandizer - April 4, 2018 at 10:35 pm
RE: Studying Mathematics Thread - by polymath257 - April 5, 2018 at 7:41 am
RE: Studying Mathematics Thread - by robvalue - April 5, 2018 at 8:48 am
RE: Studying Mathematics Thread - by polymath257 - April 5, 2018 at 9:01 am
RE: Studying Mathematics Thread - by Grandizer - April 5, 2018 at 9:03 am
RE: Studying Mathematics Thread - by Grandizer - April 5, 2018 at 8:25 pm
RE: Studying Mathematics Thread - by robvalue - April 6, 2018 at 2:18 am
RE: Studying Mathematics Thread - by Grandizer - April 6, 2018 at 2:58 am
RE: Studying Mathematics Thread - by robvalue - April 6, 2018 at 4:16 am
RE: Studying Mathematics Thread - by Grandizer - April 6, 2018 at 4:22 am
RE: Studying Mathematics Thread - by polymath257 - April 6, 2018 at 7:53 am
RE: Studying Mathematics Thread - by robvalue - April 9, 2018 at 12:58 pm
RE: Studying Mathematics Thread - by Grandizer - April 9, 2018 at 11:46 am
RE: Studying Mathematics Thread - by Silver - April 9, 2018 at 1:01 pm
RE: Studying Mathematics Thread - by robvalue - April 9, 2018 at 1:09 pm
RE: Studying Mathematics Thread - by polymath257 - April 9, 2018 at 1:20 pm
RE: Studying Mathematics Thread - by robvalue - April 9, 2018 at 1:22 pm
RE: Studying Mathematics Thread - by Grandizer - April 14, 2018 at 2:07 am
RE: Studying Mathematics Thread - by polymath257 - April 14, 2018 at 10:24 am
RE: Studying Mathematics Thread - by robvalue - April 14, 2018 at 10:41 am
RE: Studying Mathematics Thread - by Grandizer - April 15, 2018 at 5:18 am
RE: Studying Mathematics Thread - by Jarrey - April 15, 2018 at 7:12 am
RE: Studying Mathematics Thread - by robvalue - April 15, 2018 at 7:53 am
RE: Studying Mathematics Thread - by Grandizer - April 15, 2018 at 9:21 am
RE: Studying Mathematics Thread - by polymath257 - April 15, 2018 at 10:26 am
RE: Studying Mathematics Thread - by Grandizer - April 19, 2018 at 10:39 pm
RE: Studying Mathematics Thread - by robvalue - April 21, 2018 at 3:44 am
RE: Studying Mathematics Thread - by Edwardo Piet - April 21, 2018 at 10:09 am
RE: Studying Mathematics Thread - by Grandizer - April 23, 2018 at 5:51 pm
RE: Studying Mathematics Thread - by polymath257 - April 23, 2018 at 10:09 pm
RE: Studying Mathematics Thread - by Divinity - April 23, 2018 at 7:19 pm
RE: Studying Mathematics Thread - by Grandizer - April 23, 2018 at 8:32 pm
RE: Studying Mathematics Thread - by Divinity - April 23, 2018 at 10:45 pm
RE: Studying Mathematics Thread - by Grandizer - April 24, 2018 at 2:18 am
RE: Studying Mathematics Thread - by polymath257 - April 24, 2018 at 8:13 am
RE: Studying Mathematics Thread - by Fireball - April 23, 2018 at 8:20 pm
RE: Studying Mathematics Thread - by Grandizer - April 24, 2018 at 4:43 am
RE: Studying Mathematics Thread - by polymath257 - April 24, 2018 at 8:24 am
RE: Studying Mathematics Thread - by robvalue - April 24, 2018 at 5:09 am
RE: Studying Mathematics Thread - by Grandizer - April 24, 2018 at 8:18 am
RE: Studying Mathematics Thread - by polymath257 - April 24, 2018 at 11:24 am
RE: Studying Mathematics Thread - by polymath257 - April 26, 2018 at 7:48 am
RE: Studying Mathematics Thread - by Grandizer - April 27, 2018 at 7:19 pm
RE: Studying Mathematics Thread - by polymath257 - April 27, 2018 at 8:53 pm
RE: Studying Mathematics Thread - by Grandizer - April 28, 2018 at 1:41 am
RE: Studying Mathematics Thread - by polymath257 - April 28, 2018 at 9:51 am
RE: Studying Mathematics Thread - by Grandizer - April 28, 2018 at 11:35 pm
RE: Studying Mathematics Thread - by polymath257 - April 29, 2018 at 9:00 am
RE: Studying Mathematics Thread - by Grandizer - April 30, 2018 at 9:48 am
RE: Studying Mathematics Thread - by polymath257 - April 30, 2018 at 1:15 pm
RE: Studying Mathematics Thread - by Grandizer - April 30, 2018 at 7:42 pm
RE: Studying Mathematics Thread - by polymath257 - April 30, 2018 at 8:09 pm
RE: Studying Mathematics Thread - by Grandizer - May 1, 2018 at 2:19 am
RE: Studying Mathematics Thread - by Grandizer - April 28, 2018 at 9:29 am
RE: Studying Mathematics Thread - by polymath257 - May 1, 2018 at 7:55 am
RE: Studying Mathematics Thread - by Grandizer - May 1, 2018 at 7:59 am
RE: Studying Mathematics Thread - by Fireball - May 1, 2018 at 11:43 pm
RE: Studying Mathematics Thread - by Grandizer - May 2, 2018 at 6:35 am
RE: Studying Mathematics Thread - by Fireball - May 2, 2018 at 11:43 am
RE: Studying Mathematics Thread - by polymath257 - May 4, 2018 at 8:14 am
RE: Studying Mathematics Thread - by Fireball - June 15, 2018 at 12:35 pm
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - June 15, 2018 at 11:45 am
RE: Studying Mathematics Thread - by polymath257 - June 15, 2018 at 3:46 pm
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - July 6, 2018 at 3:23 am
RE: Studying Mathematics Thread - by polymath257 - July 6, 2018 at 8:42 pm
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - July 17, 2018 at 2:02 am
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - September 27, 2018 at 1:59 am
RE: Studying Mathematics Thread - by The Valkyrie - September 27, 2018 at 2:01 am
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - September 27, 2018 at 2:10 am
RE: Studying Mathematics Thread - by polymath257 - September 27, 2018 at 8:07 am
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - September 27, 2018 at 7:13 pm
RE: Studying Mathematics Thread - by Grandizer - September 27, 2018 at 7:20 pm
RE: Studying Mathematics Thread - by polymath257 - September 27, 2018 at 9:06 pm
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - September 28, 2018 at 5:34 pm
RE: Studying Mathematics Thread - by Aliza - September 28, 2018 at 6:57 pm
RE: Studying Mathematics Thread - by polymath257 - September 29, 2018 at 9:16 am
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - September 29, 2018 at 10:50 am
RE: Studying Mathematics Thread - by polymath257 - September 29, 2018 at 6:02 pm
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - September 29, 2018 at 10:01 pm
RE: Studying Mathematics Thread - by Aliza - September 30, 2018 at 2:40 pm
RE: Studying Mathematics Thread - by Aliza - September 29, 2018 at 6:21 pm
RE: Studying Mathematics Thread - by Tiberius - September 27, 2018 at 2:11 am
RE: Studying Mathematics Thread - by Aliza - September 27, 2018 at 2:23 am
RE: Studying Mathematics Thread - by robvalue - September 28, 2018 at 1:35 am
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - September 28, 2018 at 8:02 pm
RE: Studying Mathematics Thread - by Aliza - September 28, 2018 at 8:21 pm
RE: Studying Mathematics Thread - by Fireball - September 29, 2018 at 7:56 pm
RE: Studying Mathematics Thread - by Aliza - September 29, 2018 at 8:05 pm
RE: Studying Mathematics Thread - by Fireball - September 29, 2018 at 9:08 pm
RE: Studying Mathematics Thread - by Aliza - September 30, 2018 at 12:12 am
RE: Studying Mathematics Thread - by Fireball - September 30, 2018 at 7:56 pm
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - September 30, 2018 at 10:11 pm
RE: Studying Mathematics Thread - by Aliza - September 30, 2018 at 10:18 pm
RE: Studying Mathematics Thread - by Fireball - September 30, 2018 at 10:44 pm
RE: Studying Mathematics Thread - by polymath257 - October 1, 2018 at 8:06 am
RE: Studying Mathematics Thread - by LastPoet - October 2, 2018 at 7:40 am
RE: Studying Mathematics Thread - by polymath257 - October 2, 2018 at 7:58 am
RE: Studying Mathematics Thread - by Reltzik - October 2, 2018 at 10:49 am
RE: Studying Mathematics Thread - by Reltzik - October 2, 2018 at 10:58 am
RE: Studying Mathematics Thread - by polymath257 - October 2, 2018 at 11:34 am
RE: Studying Mathematics Thread - by Grandizer - October 2, 2018 at 11:05 am
RE: Studying Mathematics Thread - by Aliza - October 2, 2018 at 11:29 am
RE: Studying Mathematics Thread - by Grandizer - October 2, 2018 at 11:35 am
RE: Studying Mathematics Thread - by polymath257 - October 2, 2018 at 12:03 pm
RE: Studying Mathematics Thread - by Aliza - October 2, 2018 at 12:50 pm
RE: Studying Mathematics Thread - by Grandizer - October 2, 2018 at 1:23 pm
RE: Studying Mathematics Thread - by polymath257 - October 2, 2018 at 1:29 pm
RE: Studying Mathematics Thread - by Aliza - October 3, 2018 at 1:53 am
RE: Studying Mathematics Thread - by polymath257 - October 3, 2018 at 8:30 am
RE: Studying Mathematics Thread - by Fireball - October 3, 2018 at 11:30 am
RE: Studying Mathematics Thread - by polymath257 - October 3, 2018 at 12:25 pm
RE: Studying Mathematics Thread - by Fireball - October 3, 2018 at 1:03 pm
RE: Studying Mathematics Thread - by robvalue - October 1, 2018 at 1:54 am
RE: Studying Mathematics Thread - by Reltzik - October 2, 2018 at 2:51 am
RE: Studying Mathematics Thread - by Grandizer - October 5, 2018 at 1:19 pm
RE: Studying Mathematics Thread - by Fireball - October 9, 2018 at 8:03 pm
RE: Studying Mathematics Thread - by Grandizer - October 10, 2018 at 12:13 am
RE: Studying Mathematics Thread - by Aliza - October 10, 2018 at 3:10 pm
RE: Studying Mathematics Thread - by Grandizer - October 10, 2018 at 11:34 pm
RE: Studying Mathematics Thread - by Reltzik - October 11, 2018 at 12:20 am
RE: Studying Mathematics Thread - by Aliza - October 11, 2018 at 12:30 am
RE: Studying Mathematics Thread - by Reltzik - October 11, 2018 at 12:35 am
RE: Studying Mathematics Thread - by Aliza - October 11, 2018 at 12:53 am
RE: Studying Mathematics Thread - by Aliza - October 11, 2018 at 12:41 pm
RE: Studying Mathematics Thread - by Reltzik - October 11, 2018 at 2:47 pm
RE: Studying Mathematics Thread - by DodosAreDead - October 11, 2018 at 12:21 am
RE: Studying Mathematics Thread - by EgoDeath - October 9, 2018 at 4:20 pm
RE: Studying Mathematics Thread - by Reltzik - October 9, 2018 at 7:55 pm
RE: Studying Mathematics Thread - by EgoDeath - October 10, 2018 at 9:46 am
RE: Studying Mathematics Thread - by Reltzik - October 10, 2018 at 3:37 pm
RE: Studying Mathematics Thread - by EgoDeath - October 10, 2018 at 11:29 pm
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - October 10, 2018 at 12:07 am
RE: Studying Mathematics Thread - by Fireball - October 11, 2018 at 2:28 am
RE: Studying Mathematics Thread - by Aliza - October 11, 2018 at 10:08 am
RE: Studying Mathematics Thread - by Grandizer - October 11, 2018 at 7:31 pm
RE: Studying Mathematics Thread - by Fireball - October 11, 2018 at 8:34 pm
RE: Studying Mathematics Thread - by Reltzik - October 12, 2018 at 11:34 pm
RE: Studying Mathematics Thread - by Aliza - October 13, 2018 at 2:07 am
RE: Studying Mathematics Thread - by Kernel Sohcahtoa - October 13, 2018 at 6:40 pm
RE: Studying Mathematics Thread - by Fireball - October 13, 2018 at 8:15 pm
RE: Studying Mathematics Thread - by polymath257 - October 14, 2018 at 11:33 pm
RE: Studying Mathematics Thread - by Grandizer - October 27, 2018 at 11:39 am
RE: Studying Mathematics Thread - by Fireball - October 27, 2018 at 11:54 am
RE: Studying Mathematics Thread - by Aliza - October 28, 2018 at 12:23 am
RE: Studying Mathematics Thread - by Grandizer - October 28, 2018 at 7:42 pm
RE: Studying Mathematics Thread - by Silver - October 30, 2018 at 6:03 am
RE: Studying Mathematics Thread - by Aliza - October 30, 2018 at 10:55 am
RE: Studying Mathematics Thread - by Grandizer - October 31, 2018 at 10:14 am
RE: Studying Mathematics Thread - by Grandizer - November 3, 2018 at 1:15 am
RE: Studying Mathematics Thread - by Grandizer - November 3, 2018 at 6:30 am
RE: Studying Mathematics Thread - by Fireball - October 30, 2018 at 6:44 am
RE: Studying Mathematics Thread - by Jehanne - November 2, 2018 at 9:02 am
RE: Studying Mathematics Thread - by Macoleco - November 3, 2018 at 11:15 pm
RE: Studying Mathematics Thread - by polymath257 - November 4, 2018 at 9:26 am
RE: Studying Mathematics Thread - by Jehanne - November 5, 2018 at 9:55 am
RE: Studying Mathematics Thread - by polymath257 - November 5, 2018 at 1:03 pm
RE: Studying Mathematics Thread - by Jehanne - November 6, 2018 at 8:37 am
RE: Studying Mathematics Thread - by polymath257 - November 7, 2018 at 1:31 pm
RE: Studying Mathematics Thread - by Jehanne - November 7, 2018 at 3:51 pm
RE: Studying Mathematics Thread - by polymath257 - November 7, 2018 at 5:30 pm
RE: Studying Mathematics Thread - by Jehanne - November 8, 2018 at 11:34 pm
RE: Studying Mathematics Thread - by polymath257 - November 9, 2018 at 10:20 am
RE: Studying Mathematics Thread - by Silver - November 8, 2018 at 10:22 am
RE: Studying Mathematics Thread - by bennyboy - November 19, 2018 at 12:48 am
RE: Studying Mathematics Thread - by Fireball - November 19, 2018 at 1:28 am

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