Metal Integral
Вставка
- Опубліковано 8 вер 2024
- In this video, I calculate the integral from 0 to n of x/2sqrt(x^2+4) + n+2/2n using trig substitution. The result is called the metallic ratio, where the case n = 1 leads to a number you probably all know :) Enjoy!
Note: This problem can also be done using a regular substitution, but trig substitution is cooler 😎
Trisub seems to be an overkill... Btw thx for ur effort in producing math vid
Maybe I missed something, but could we just do u-sub with u=x^2+4 and then the integral just follows the power rule?
He does say this in the description!
Here is a quick proof writeup www.overleaf.com/read/rwkchcdxnksx if anyone is interested, but it's not very hard.
Ye
Did this too, way easier
Yeah, but the method in this video is more metal 😎
not even that, just let u=\sqrt{x^2+4}
I most certainly enjoyed this little trig integral calculus metal excursion! The DR ROCKS the house!
The Integral is actually beautiful..
Golden,Silver and Bronze Ratio....thats all together Metalic Ratio...COOL!!
1, 2, 3, 5, 8 and 13 are all Fibonacci numbers ...
I enjoy watching your videos, each time it's a pleasure
Awesome. I won't claim x^2+4 sub is easier since it was already done. Instead, because I don't like Metallica, I'll wait for the Iron ratio. Up the Irons!
13+sqrt(170)?
The "metallic" result is very cool. I do get the same result with a u-sub. Thanx for producing the video.
This made my day metallic..
I also see your plastic ratio video really coooooooolllllll things are to do..
Thanks sir🙏🙏🙏🙏
Who else thinks that Dr. Pi(m) should deserve more subs?
One of my friends works at our state library and for my birthday this year he got me a series of math exercise books from 1917. One of the problems is the following integral: integrate (tan(x)*log(cot(x))) from 0 to pi/4;
This turned out to be way harder than I thought and i'd really like to see you solve it :D
It would probably help if that integral actually converged, as wolfram alpha really doesn't think it does.
www.wolframalpha.com/input/?i=integrate+(tan(x)*ln(cot(x)))+from+0+to+pi%2F2
@@hOREP245 whoops my bad, it's from 0 to pi/4
fNktn Sub t=tanx then use geometric expansion of 1/(1+t^2)
Gracias Dr Tigre Peyan, saludos desde Bolivia en su aniversario !!!!
Anyone else solved this in their head?
Thanks for the heavy metal(s) video! Pure gold!
Wow! Originally I think metallic ratio is somewhat related to metallic bond but at the end it is related to the three cool ratios, thank you Dr Peyam!
Integral is different in the video and thumbnail by a factor of 2 :)
AMAZING! 🤘🏻😄🤘🏻
I didn't expect this would be interesting until I heard silver ratio! lol
keep making interesting videos on mathematics
What books do you recommend to learn advanced math?.
Awesome video :D
All basic textbooks in calculus are good. The favoured one is by Stewart.
It depends what you already know. It's really important to know basic stuff about sets and related notation, as well as functions. You should learn about logic and different types of statements, e.g. contrapositive, existential and universal statements. This includes different techniques of proof, such as induction and contradiction. Working knowledge of these concepts aids while writing and reading proofs, which is incredibly common in math as you go further on. All of these things, I would consider core to studying 'advanced math.' So if you haven't studied them yet, you should start there. Thankfully, there are lots of books on these topics, some free and accessible online-you could probably find lots of suggestions on reddit or other websites. Popular book suggestions for these topics are Velleman's "How to Prove It" and Hammock's "Book of Proof." There are other books that would likely serve you just as well, though those are nice from what I've read of them.
After you have these preliminaries, there are multiple routes you could take: My advice would be to pick one thing and do it well. It is easy, in my experience, to overextend oneself while trying to learn lots of new information. And sometimes that's fun; it can also result in feeling lost, and if you don't have someone more experienced to help you along, it can be tough. Calculus may already be partially familiar, so it might make sense to start there. An 'advanced calculus' book would likely expose you to more rigorous math while still drawing on what's somewhat familiar, which is nice. Sometimes (not always!) rigor can obscure why something is true, so having the existing motivation is helpful. If you wanted to take a different route and study, say, abstract algebra, lots or all of the stuff could seem unfamiliar-that's fine too, it's just a different adventure at first. If you end up studying calculus/analysis, there'll be lots of people recommending a book by Rudin (Principles of Mathematical Analysis). This is a fine book, but it's pretty intense, and there are many other analysis books to choose from that might be more illuminating to start off.
Books I'm enjoying (but admittedly very slowly) are Sheldon Axler's Linear Algebra Done Right and Steven Abott's Understanding Analysis. If you try these books get ready to spend at least an hour per page.
I recommend you books from Tom Apostol and Piskunov, they could be complicated at first time because the Calculus' theory. But also they're very interesting;)
Don't know why
🤘 I understood this one properly ☺️😋
Somewhat exotic to see (co)secants used. Never needed them, myself, plus they're kind of weird. sine and cosine are 'nice', but their reciprocals are not.
this is so cool!!!
You can easily do it and you don't have to use substitution method
Integral x/(2sqrt(x^2+4)) = sqrt(x^2+4)/2
Because the d(sqrt(x^2+4))/dx=
x/(sqrt(x^2+4)).
Power rule & chain rule effortlessly
Cool :)
Amazing videos as always!! Side note: What source would you suggest for learning lie algebra? Thanks a lot
Its easier to subsitute sinh because 1 + sinh^2=cosh^2
And now the stone- and wood-integral ^^
Hahaha
@@drpeyam
ln(x)+ln(x-1)=0 --> x = golden ratio
ln(x)+ln(x-2)=0 --> x = silver ratio
ln(x)+ln(x-3)=0 --> x = bronze ratio
...
So: phi_n = n+1/(n+1/(n+1/(...)))
I've noticed something neat with the metallic ratio and the series a(n) = n^2 + 1 (oeis.org/A002522).
When you calculate Phi(n) when n is even, you get this :
Phi(2) = 1 + sqrt(2)
Phi(4) = 2 + sqrt(5)
Phi(6) = 3 + sqrt(10)
Phi(8) = 4 + sqrt(17)
Phi(10) = 5 + sqrt(26)
Phi(12) = 6 + sqrt(37)
And so on...
Then when you look at the series "a(n) = n^2 + 1" and let a(n) be the number before the + sign, you get the number inside the square root.
I wonder if there exists an equivalent relation to a series when Phi(n) is odd ?
Cool fact: if you plot y = (n+√(n^2+4)) / 2 as a function of n, as n gets bigger and bigger, the graph asymptotically approaches y = n
Desmos link: www.desmos.com/calculator/ux7q1cjqur
If you talk on metal, do not forget to mention plastic by the way. : )
Hi @Dr Peyam! I wanted to know if it was possible to decompose the exponential function into a Fourier serie in the real world (cause otherwise I know tha exp(ix) =cos(x) +i sin(x)
Wait maybe I didn't understand a thing about fourrier series
The silver ratio is interesting because (√2 +1)^n as n→∞ approaches an integer.
i know metallic ratios defined analogously to the golden ratio, as a proportion of special rectangles. Is this integral just a coincidence, or is there some significance to it?
That integral screams for a u substitution. Dr. Peyam is unphased and uses trig substitution instead.
It screamed trig sub for me!
The substitucion tg is useless. We can use the substitution k= sqrt(sqr x+4)
Miss'd the "2s" in the Denominator in the Thumb-nail, Sir.
Use of parenthesis, will make it mathematicaly fair ;-)
triggggg
What about diamond ratio? ..
Presumably the Platinum Ratio is 1?
Oh, and definitely screaming u=x^2+4 sub, what with the du already on top.
_Bronze ratio_
Alloy there!
That confused me, too. And what follows after Bronze (e.g. N=4)?
@@weinihao3632
Well, the atomic number of beryllium is 4, but they can't really name them that way - especially as atomic numbers 1 and 2 - hydrogen and helium - aren't normally metals.
Group 11 of the periodic table might suggest that, after gold and silver, copper should be next (I actually expected that).
Perhaps the numbering might better have been called coinage or medal numbers.
Cheers :)
How about metallic ration with N being negative
Then you get the unobtainium, adamantium and mithril ratios.
Was expecting some nice cheeky graph that looked like Metallica, got baited by Thumbnail, was disappointed.
Why are US pronounciations of greek letters so triggering? Theta rhymes with "metre" and phi rhymes with "why". After all, you don't call 3.1415... "pee", right?
The US went to the moon with those pronunciations. It works, even if they printed everything upside down and read it backwards. :)
But,do you actually like Metallica?
I don’t really listen to them
Dont do another video about properties of metalic ratios. There are already enough of those on UA-cam
Oops!
@@drpeyam this video was quite different, but there are a lot of videos with typical properties of the metalic ratios, so I urge against making one, unless you can make the video about different things
Haha, I already prerecorded one 😅 Let’s see how it goes