a geometric approach to a famous integral

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Опубликовано:

15 сен 2024

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Комментарии : 64

@tommasoantonelli7176 4 месяца назад

9:32 -> Ridiculous place to stop 20:03 -> Good place to stop

@edwardfyodorov8268 4 месяца назад

This is hilarious

@Happy_Abe 4 месяца назад

Love this 😆

@redpepper74 4 месяца назад

Exactly my thoughts

@Ahmed-Youcef1959 4 месяца назад

👍👍👍

@williamperez-hernandez3968 4 месяца назад

The slope of the line is 1/t, so we get x=t/sqrt(t^2 +1). This does give y=1/sqrt(t^2+1) -1 as given in the vid.

@nahuelcaruso 4 месяца назад

Yes, the misleading slope t leads to a lower bound equal to (t/sqrt(t^2+1) -1). However, with this little correction the proof runs without problem

@JeanYvesBouguet 4 месяца назад

The best part of this video is when showing the equality of areas between the As and the Bs respectively. This is the most interesting and non intuitive part of the method in my opinion.

@hydropage2855 4 месяца назад

Several mistakes starting at around 13:00. The slope was supposed to be 1/t. You forgot the square root when solving for x, and even though you remembered it later you forgot the t when plugging into y = tx - 1. Should’ve been t/sqrt(t^2 + 1) - 1. You got lucky and your mistakes canceled out

@djttv 4 месяца назад

Who would have thought that A1=B1 and A2=B2? Very interesting video!

@yutaj5296 4 месяца назад

13:05 The equation of the line should be y=x/t-1. The result that y=1/sqrt(t^2+1)-1 is correct.

@Jack_Callcott_AU 4 месяца назад

👍 You are correct!

@Jack_Callcott_AU 4 месяца назад

It seems that the mistake he made cancels out. ✔

@Milan_Openfeint 4 месяца назад

Now we need a geometric argument why A1=B1 and A2=B2.

@MathTutor1 4 месяца назад

This is beautiful. Keep up the good work.

@goodplacetostop2973 4 месяца назад

20:03

@BrianDominy 4 месяца назад

9:31 Not a good place to stop

@gp-ht7ug 4 месяца назад

Cool video! I like when you put together geometry and calculus

@四步道君 4 месяца назад

and 1/(1+x2)=(i/2(x+i))-(i/2(x-i)),so tanx=(i/2)ln((x+i)/(x-i))+C.You can find the value of C by taking the derivative of sine x or cosine x.This is how most civilizations in the universe connect the real and complex domains:)

@bsmith6276 4 месяца назад

When introducing the substitution u=sqrt(2y)-1 I think if you broke up the integrand into sqrt(1-2y) / sqrt(2y) then the substitution may seem a bit more motivated since dy/sqrt(2y) is the differential of sqrt(2y).

@estudematematica 4 месяца назад

Yet another great video, but I have a feeling that it got unnecessarily rushed from minute 15 or so… we’ll be around if it takes an extra minute or two, Mr. Penn! 😃👍

@Bruno-yg9lu 4 месяца назад

@DeJay7 4 месяца назад

Never explained the reasoning behind the 1/2 factor in the initial function, very strange. I think it just happens to make Area(A1) = Area(B1) and Area(A2) = Area(B2) instead of having a factor of 2.

@manucitomx 4 месяца назад

This was great fun. Thank you, professor.

@coreymonsta7505 4 месяца назад

That’s the bigger picture of FTC1,2! Anti derivatives are like cumulative area functions, since they’re of that form up to a constant

@bethhentges 4 месяца назад

13:48 He means x^2=

@The_Green_Man_OAP 4 месяца назад

13:47 Mistake here. Should be x=1/√(t²+1)

@ryoikitokuiten 4 месяца назад

Wow. Really nice approach.

@backyard282 4 месяца назад

only thing i didn't quite understand is why you you were using the function (1/2) * 1/(x^2+1). Why the 1/2?

@Aditya_196 4 месяца назад

For the last part the way we calculated the arcs area we got the 1/2(theta)*1² ..if u won't do that u will still get same sort of stuff but I believe the people who come up with the proof already had the basis of proof as using that arcs area so they just did it with 1/2 factor in the beginning to avoid some extra computational efforts 👍🏻

@Jack_Callcott_AU 4 месяца назад

Hey @backyard282, I think the reason for that is that 1/( X ^2 + 1) is an even function, so the integral from -t to t on (-t, t) is 2 times the integral from 0 to t on (0, t). ✔

@bsmith6276 4 месяца назад

Probably because he can then say area A1=area B2 and area A2=area B1 near the end.

@maurobraunstein9497 4 месяца назад

I just saw another video about that this week! ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-ZcbWjD6HA_s.html Another Roof's geometric interpretation is quite different, drawing a circle under the Witch and comparing the area of a slice of the Witch to the area of a slice of the circle. In the more than 20 years I've known that the integral ∫dx/(1 + x^2) is arctan(x) + C, I had never, until a few days ago, even considered that there might be a relatively simple geometric interpretation, and now I know two!

@ingiford175 4 месяца назад

It is seems to be the same general proof, but the location of the circle is in a different place. Penn has his circle under the x axis, while the other proof is a circle half the radius within the the curve and the x axis. Both are interesting and forgot I saw the other version earlier this month.

@purplerpenguin 3 месяца назад

This integral proof is so complicated that I just don't see it as having merit. The standard trigonometric substitution is so simple.

@CTJ2619 4 месяца назад

well done - i liked the visual representation of what was going on -

@黃逸驄 25 дней назад

Personally I think it'd save some time and effort at 12:40 by spotting a pair of similar triangles rather than solving simultaneous equations

@byronwatkins2565 4 месяца назад

At 14:00, tx-1=t/sqrt(t^2+1)-1

@bellfoozwell 4 месяца назад

Nice explanation!

@jounik 4 месяца назад

Why was it necessary to even split the area below the x-axis? It's a triangle with base t and height 1, so its area is t/2. It would've been easier - and less error-prone - to just complete the integral for A1 and show that A1 and A2 sum to t/2.

@Detka48 4 месяца назад

Because the whole point is to avoid calculating any integrals.

@erfanmohagheghian707 3 месяца назад

You had all the coordinates of the intersections. Why did you use similar triangles? :)))

@pieters286 4 месяца назад

most enjoyable derivation!

@bethhentges 4 месяца назад

15:40 He wrote the similarity in the wrong order.

@nicolascamargo8339 4 месяца назад

Wow increible

@udic01 4 месяца назад

Like everyone else commented, the slope is 1/t

@zh84 4 месяца назад

Fascinating. I never would have thought of this, but you led me right through it. Thank you Back in 1991 I tried expanding exp(-x²) with Mathematica in powers of 1/(1+x²) because I thought the two functions looked similar (both equal to 1 when x = 0, both asymptotic to the x-axis) but it turned out to be horribly messy.

@klofat 4 месяца назад

Where is +c? Great video, enjoyed it a lot.

@levprotter1231 4 месяца назад

Any hyperbolic equivalent?

@marc-andredesrosiers523 4 месяца назад

good job 🙂

@letitiabeausoleil4025 4 месяца назад

I only get abouyt 1/50 of your problems out before the video ends. Behold! This was one of them.

@kruksog 4 месяца назад

Why introduce the factor of 1/2?

@bethhentges 4 месяца назад

Because it appears at the end when finding the area of the sector.

@kruksog 4 месяца назад

@@bethhentges don't you think it might be a good idea to explain or say something about that, rather than allowing it to be an utter surprise at the end of a nearly half hour derivation?

@easymathematik 4 месяца назад

Nice topic.

@Aditya_196 4 месяца назад

I have wondered it for a long time if he would rename us channel to michael pen²

@MacHooolahan 4 месяца назад

Is there a (different) proof here that involves d-theta-ing some angle round from the origin to intersect with that pythag-y function line? Feels like there is but it's probably not going to come from me right now, having had several beers! :O