Innocent looking, but ????
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- čas přidán 2. 10. 2018
- This is an innocent-looking integral but it's actually dangerous. The integral of 1/x^2 from -2 to 1 is a type 2 improper integral because it has a vertical asymptote on the interval of integration. This improper integral actually diverges! Be careful with the criteria when we use the fundamental theorem of calculus part 2. #calculus #apcalculus #blackpenredpen
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"I'm tired of doing my math homeworks. I'm going to watch a video to chill a little"
same here😂😂😂
Top 10 things said before disaster
This video woke AF
Me rn
Did I ask
I can't believe I just watched a 10 minute video on integrating a function.
And I enjoyed every minute of it
Me too. I'm not even in calculus anymore :D
Yes, but why talk so much around this?
SAME
Its 7 am and this video made me want to do the maths by myself
@@mangoface7914 it’s 7am here too as I watch this
History repeats itself
@@MrAlsPals i have another exam tomorrow but I'm watching this and doing math lolllllllll
There is a prof in my Uni, he gave us a function in Arithmetic Analysis and said "I am going for a cig, solve it", when he came back he asked, "did you guys solve it", some said yes, and then he said "Too bad, there is no analytical solution, the function is not linear". He trolled us hard
flex move
Then how the hell did you guys manage to solve it?
@@createyourownfuture5410 We didn't, I think he solved it using the taylor series. It's been almost six years since then
@@asterdogma but how did some said yes then?
@@createyourownfuture5410 because most of the students in my Uni thought that 1 byte equals 4 bit and most of them calculate probability to be 2.9. You get the idea, they thought they had it solved because they don't know what linearity is
1:15 When I found out steel is not heavier than feathers
1 pound of steel vs. 2 pounds of feathers : )
@@blackpenredpen lol the 2 pounds of feathers
I understood that reference
Subin Manandhar Good :)
I read somewhere that 1 ton of feathers is actually heavier than 1 ton of bricks because you also have to deal with the guilt of what you did to those poor birds.
“Sometimes when you have an innocent-looking thing it’s actually evil”
man, that can be applied in many ways
Like?
@@createyourownfuture5410 women
@@MultiChrisjb oh
@@createyourownfuture5410 people.
@@MultiChrisjbwhat about men?
1:45 My AHAA moment, when I realized that 1/x^2 is always positive, so how can the integral be negative?
[insights]
: )
If you integrate "backwards" on a function, you can get negative values. Like integral from 0 to 1 of x^2 is 1/3 but integral from 1 to 0 is -1/3
Integral values can be -'ve if it is below the graph. But you take the modulus of it when you have the final value.
Every time integration don't define the area...in different scenario it defines something else also
Um, then... can we consider -3/2 as an anaylitic continuation of integ [x=-2 to 1] x^(-2) dx? Actually that integral diverges, but we can get -3/2 with invalid way, so if we give meaning to that value, I think we can consider -3/2 as an analytic continuation. :)
1:31 The best lesson of my life. Thanks
Children in a nutshell
Womens in general.
Why am I watching this at 1am I'm not even in calculus
dont worry, you are not alone on this one
Wth am I watching this.. I haven't even got to trigonometry yet XD (but I know how to do basic trig)
in which grade to you do calculus?
Crystxllize in the states. Some states let you do it in 12th grade if you were selected in 8th grade to skip 8th grade math and go straight to 9th grade math (algebra 1). But for 99% of US students, it will be in college
@@maxhill504 wtf, in Germany everyone does it in 11th grade
“This is what you say to your girlfriend...”
Every math nerd watching: *visible confusion*
@Lo Po I don't think there's anything wrong with being a nerd either ......
@Lo Po But being dumb and ugly does, sadlife
@Lo Po You are asking me what my point is, but then give your opinion about the point I didn't make???
oh hey, not bragging or anything but I'm a TOTAL math nerd and I have girlfriend! uwu
@@adamantmist9394 hey not shooting my self in the foot but it seems I have shot myself in the foot. oowoo
Jeff Bezos is not going to be happy with this video.
: )
What are you referencing?
@@geico105 08:52
なんで?
jeff pesos XD
if you want to make the integral work for practical purposes then add a small number to x^2 e.g. 1/(x^2+0.0001)
Then it still runs past /0. Just as an offset for whatever constant you put?
@@unclebenz86 How can (x^2 + 0.001) = 0 for real x?
@@Raddaya solve x^2+0.00001 for x? That would be 0 for a certainly Set oder numbers.
@@unclebenz86 there is no real number solution to that
x^2 = [any negative integer] has no real solution, such a denominator can't be 0
The original comment said that it was to make it work for practical purposes
You can make the integral arbitrarily big by making that small number smaller.
This is great! Clear explanation and warns us about the pitfalls of points where a function is discontinuous.
I thought it was clickbait when he left the board
Literally learned this yesterday and I found this very entertaining. Thank you for a good example and not a problem set up to lead you to a specific answer. That problems shows why you have to be careful integrating, and I really enjoyed learning from your video
This is a great example to remind students that although performing FTC calculations is important, it's much more important to understand when we can do this and why and what to do when we can't. This teaching creates thinkers not machines. Bravo. bprp
you make me love maths, I really enjoy the vibes you give to these exercices, thanks for being on this planet mate
1:21 the evil laugh your calc professor gives you when correcting your answer
Bruh, I plugged this into my calculator (it can do integrals) and it freaking crashed 😂😂😂
😂😂😂😂😂😂😂
Same dude lol
9:00 J. Besos:
TRIGGERED! Don't take away my millions!
Okay, do the same thing but with 1/x^3. Since it is an odd function, you can use symmetry to cancel out the diverging parts. Would be fun!
I have a similar one, czcams.com/video/dHwrzLDmdT8/video.html
You could use ppoam to the power of b=″€¥∆¶\([®™©Ωπ•
You could have just done a quick convergence test on 1/X^2 or on the indefinite integral before trying to split up and evaluate the integral. If the original function diverges, then so does an integral of the original function, or if the integral diverges at one of the points of discontinuity.
"If the original function diverges, then so does an integral of the original function" - Not true, buddy. The function
f(x)=1/|x|^(1/2) diverges at 0, but the integral of f(x) between -1 and 1 is a finite number.
@@presorchasm: Why would you say the integral is zero? In general, if the integrand is non negative and >0 on a positive measure set, then the integral will be strictly positive (possibly infinite).
@@rv706 nevermind lol, I had the wrong computation
i like how you explain these things with a smile. :)
This video is so great to help me clarify my concept.Thank you
Excellent presentation of the topics in a beautiful manner . Vow !
I love watching your videos, Steve! You have taught me more than my teachers ever did. And that was 25 years ago! Keep up the good work 👍
Thank you very much Mike!
So unexpected, but yet so logical. It always amazes me how math always makes sense.
Wow that is really interesting. These videos are great, thanks for your hard work. Keep it up man.
Thank you a lot! I needed this refresher!
Thanks for all your videos mate, I really love to watch you do maths! Must admit, as soon as you found the area under the curve, I quickly checked the graph on another site and spotted that might be a tiny little problem around x=0 .... really well explained though. Need to look into Limits a bit more.
Stefan McNamara : )
You're welcome. I am glad to hear that you like my videos.
The Doraemon theme playing in the background 😂😂😂
Thanks for making these example videos!
very nice one! I realised what was the catch right after you boxed the result.
I love examples like this, you done another good job ;)
Shouldn't you change the channel name to RGBpen?
😂
That's a nice name tho....thanks....i was in search for a similar name
Love your videos! Great calc 2 refresher!
your channel is amazing. thx for sharing ur knowledge
Thank you!
"Is this an easy problem?"
"Well yes, but actually no"
actually yes.
Me: *Does calculation* Ah so the answer is -3/2
Also me: *Looks at the question again* Wait a minute..... This is devil's trap in integration.
🤣🤣 so true
Excellent presentation! I love your explanations.
I'm basically at the end of my Calc I class, and I just learned so much from this video. I can't wait for Calc II!
"When you have a innocent-looking thing, it's actually pretty evil, so be careful" Blackpenredpen
It is always a good practice to define the interval of the function first. So if one follow the flow properly, there is no worry about such mistake :)
I've been liking your videos to the point where these have become a means of procrastination for me.
all of your videos are so great!!
Thank you
I love this example. Too many students walk through calculus plugging in formulas and don't take a second look at what they actually are doing.
Minor point but It seems reasonable for the integral from -2 to 1 of 1/x^2 to say that it “diverges toward infinity” since both improper parts have a limit that is unbounded toward infinity. That would be to distinguish it from the same integral for 1/x or 1/x^3 for example where the integral could instead be called “indeterminate” since those have divergences in opposite directions. It seems like a handy distinction since knowing if the divergence is toward positive or negative infinity versus being truly indeterminate tells you something useful about the actual behavior of the integral and function around the point of discontinuity.
Thanks for helping me fall in love with maths again.
Thanks dude, this really helped :D
did this wrong on a test this week :(. press f
F
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Damn bruh you are awesome in explanation
Thank you for such a beautiful question with solution.
I used to watch your videos for fun even though I never understood them because is as in year 11. Half a year later, this all makes sense and is still fun
I made this mistake on my Calc 2 final😒
This guy improved my Integration techniques😂😂😂
I just love your vids and explanations
Thanks!
Felt like watching some math videos, have my doctorate in engineering so I was thinking no watch out for that asymptote. But I really want to hear the rest of this 10 minute video just in case he mentions the name of the next step of if you get an indeterminate form after breaking it up and I can go read about it. Thanks for that last 20 seconds!
"diverges, it's a *verb*"
Never thought I'd be learning English in a math video
I feel like situations like this is why I only got a 2 on my ap cal exam, because I felt like I understood everything and can't really remember this situation coming up in class.
Except this is a very basic concept. For indefinite integrals, it's assumed that the answer is for the values of x that is in the domain. For definite integrals, the first thing to always check is that the integrand(function you're finding the derivative of) is continuous within the limits of integration. Even before your AP exam, there must have been a test that dealt with improper integrals where you were asked to find what seemed to be a trivial definite integral problem but turned out to be more complicated.
@@znhait hahahahaa ok chief 🤑😩
Thank you.You are a good teacher
So glad i found this a week before my exams
1:15 existential crisis.
I love your haircut though. 0 dislikes.
Thanks!
I don't understand who are the 72 people who disliked it? (as by 10:48 pm 11-11-2018)
It's 418 by 2:05pm 1st May 2020.
"When you have an innocent looking thing it's actually pretty evil"
My ex bro 😂😂😂
Son Goku your ex bro?
@@TheGamingKamikaze or your bro ex?
He meant my ex,bro
eX bRo?
I have a distinct feeling that I missed this on the homework I turned in yesterday :[ . Thanks so much for the informative videos!
Interesting question mate! I enjoyed it😄
Video on jee advanced problems
This same question was in my engineering entrance 😭😭
Jee?
I like it how we were just studying improper integrals earlier in class then suddenly this popped out in my yt recommendations.
I dunno this before but when i watch ur vid i understand it sooooo clearly!
You really amazed me and I amazed my teacher!!!! #yay
Soumya Chandrakar your teacher didn’t know this?
@@giancarlodisalvo1784 his teacher is probably amazed that he knew this
Top 10 greatest anime plot twists of all time.
Finally!! CZcams recommended me this I was wondering about the same for the past 6 months and of course didn't get any answers from my teachers
This is great, I haven’t seen this shit in forever but was a nice throwback to college
I think u should try probability which is considered as 1 of the most difficult topics in maths
@Cool Dude if u r in class 9or 10 then it's easy but if u r in class11 then there is nothing much difficult than probability
Brownpenbluepenblackpengreenpen
jblac201 3Brownpen1Bluepen 😂
@@johnathanwhite4878 😂😂
Great video!
The flashbacks from my calc 2 class is haunting me again
#brownpenbluepen
Yup : )
@@blackpenredpen thank you for the heart
Is the ship name for blackpenredpen and 3blue1brown?
3pen1pen
@@paytonrichards784 dude 4 pen
Great video!
Can you explain more about the Cauchy Principal? 9:45
I think the Cauchy Principal allows you to evaluate these divergent integrals by "sidestepping" around the singularities in the integral domain by going through the complex plane.
Great stuff indeed!
That’s kinda why you always need to check for division by zero, it’s almost the only thing that can destroy continuous functions other than piece wise and jump discontinuities
"Sometimes when you have an innocent looking thing it's actually pretty evil." -redpenblackpen's advice on girls
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Laughing because you must write blackpenredpen!
It's the advice for boys
If sum of numbers upto infinity can be -1/12 why cant area upto infinity be negative🤔
Well infinity is t 1/-12
The series was convergent, hence it had a definite value. On the other hand the integral is divergent.
@@abhinavshah2734 In what universe is 1+2+3... convergent? The entire -1/12 result is predicated on disregarding the radius of convergence
I can't stop watching your videos
5:50 i was waiting for this mistake coming up. In fact, u cant say ( in the case of a some of integrated or some of an infinite series) that if one part diverges then the some will diverges o , the simple example to this is the integral of 1/x from -inf to +inf if we separate the integral into two parts then they will diverges but their somme converge to 0.
Can we make integration from parts,
From -2 to 0 and from 0 to 1?
There will be a 1/0 there , than it not work
he literally did that in the video
No, because 1/0 doesn't make sense; you must use limits.
1/0 ≠ ∞
We must separate into two parts since there is a vertical asymptote at x=0. If one or both of the integral diverges, then the answer is somewhat infinite, or it diverges. If both integral coverages, then the answer must have a finite value, or it converges.
@@yafi2475 Ikr! Lmao
"know your integration, and believe in your limit"
0 : heeelllpppp It limits meeee
Really good lesson.
Amazing explanation
Just in time when I'm going to be tested on improper integrals in 6 hours. Only that all of them will have:
Determine if each of the following integrals converge or diverge. If the integral converges determine its value."
So no seemingly innocent but evil integrals.
1:22 When u get an A without studying...
0:38 - Done
1:01: you got the total area right, eh?
1:38: only works if the function is continuous?
Fundamental theorem of calculus, part 2, but there's a flaw?
2:20: Okay, I will try to understand.
3:18; recognize if it's continuous?
So it's a type-2 integral?
Brilliant . Well done.
Why did you divide by zero, you doomed us all.
What does that wrong answer -3/2 represent? I know it is wrong, but it came from a method and hence it has some meaning, but I don't get what it is.
integral computes area of the region but since it's negative on a region where its always positive it doesn't make sense (I think)
@@DOMINANTbeats No bro, integral can be negative if you are computing the area below the x-axis. Simply integrate -x from 0 to 1. The answer is negative (-½). The reason is that the graph of the curve is below x-axis when 0
Must be positive as graph over x axis and -2 less than 1.more over the function must be continuous in the interval (-2,+1).here not the case for x=0.
ward
The problem is that the function must be continuous and derivative all over the interval wich is not the case for x=0.
This is cool, just found your channel and really enjoying your videos. Lots of really interesting maths snippets taught in a way that I can understand. It is improving my understanding of maths! Keep up the amazing work
Split the integral into the following ranges - [-2,-1] and [-1,1]. Former area = 1/2 and latter is even function so twice area from [0,1]. Split into [0,h] and [h,1] where h=10^-n. Area = .5 + 2x(10^n - 1) = + infinity as you increase the power n. You can choose any n and get accurate Area A(n) as a function of n.
I want you to be my math tutor
Find me on YT : )
Does the result -3/2 carry any meaning?
No, it is nonsense. It is like saying 1+1=1 because you forget to add the 1 in the algebra.
JASS Cat that’s not a good analogy
misotanni thanks
At least I tried ok
It's like saying that 1+2+3+4+... = -1/12, even though in reality, it diverges
As soon as you said it's not continuous I was like "ah... damn it... now I remember what to do..." but then again it's been over 10 years since I did any of this stuff lol.
I love your videos