I integrated the quadratic formula
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Күн бұрынUltimate math for fun! Last year we differentiated the quadratic formula, so today we will integrate the quadratic formula! This is my first time integrating the quadratic formula and I was surprised that the answer turned out to be pretty nice! This is a super fun problem to challenge any Calculus 2 student. In the end, we will check if ChatGPT can integrate the quadratic formula for us.
Check out how to differentiate the quadratic formula: • I differentiated the q...
0:00 We will integrate the quadratic formula
7:48 ChatGPT vs integral of the quadratic formula
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Thank you all!
@blackpenredpen 10 ай бұрын
Check out how to differentiate the quadratic formula: kzfaq.info/get/bejne/gKuTeJCoqbixc58.htmlsi=Uq1p1KYNj-Kpu5g2
@KluhlHloo 9 ай бұрын
Integrating the quadratic equation = god mode! You sir are a badass!
@abishworpandit5200 9 ай бұрын
Well everything has physical meaning so what does it physically mean?
@KluhlHloo 9 ай бұрын
@@abishworpandit5200 🤔
@1Konu1Zoru 6 ай бұрын
dont rely on AI and dont ask AI to any math question.. it is very dumb in that area and still thinks that it knows the answer.. but in near future it will eventaully learn to do math and most likely beats any human being on earth :) wolfram is around for a long time and it is best tool to check results or solve something for the timebeing.
@BradleyG01 10 ай бұрын
If there’s one thing that ChatGPT tells us it’s that AI won’t be replacing math professors anytime soon.
@aashsyed1277 10 ай бұрын
Well that may not be true in the future....
@resurrectedpa 10 ай бұрын
@@aashsyed1277won't be soon 😅
@acuriousmind6217 10 ай бұрын
there's already multiple AI's online that can integrate any integratable function and give you the full meticulous steps idk if you count that as replacing math professors.
@littlegrass320 10 ай бұрын
@@acuriousmind6217 a math professor does a lot more than integrating functions, they deal with a lot more abstract concepts that don't have a set way to do them. Until those AI's can understand abstract concepts, math professors won't be going anywhere
@teelo12000 10 ай бұрын
ChatGPT write me a witty youtube comment reply.
@badamson 10 ай бұрын
If you change the last integral (1/(u^2-b^2)) with diff of two squares you can split it up with partial fractions and use natural log… although it turns out the end result is actually an identity for inverse hyperbolic tan
@Ninja20704 10 ай бұрын
They aren’t entirely the same because of the domain difference. tanh^-1 (x) is only valid for -1
@bobbyheffley4955 9 ай бұрын
@Ninja20704 if x>1 or x
@liamernst9626 9 ай бұрын
w pfp
@mrfarooqkhan8454 9 ай бұрын
@@Ninja20704bro can you explain why sometimes we get two different answers for the same integrant? How to decide which one is correct?
@Ninja20704 9 ай бұрын
@@mrfarooqkhan8454 Usually it is because the two different answers are actually just off by a constant, so they are essentially the same as far as the indefinite integral is concerned. One example is secx*tanx. In this case, the different answers we are getting is more of domain, but the functions have the same value for all x in the common domain. The integral of 1/(1-x^2) has 3 seemingly different answers: tanh^-1 (x), coth^-1 (x) and ln|(1-x)/(1+x)| tanh^-1 (x) = 1/2*ln[(1-x)/(1+x)], which is only good for -1
@CorrectHorseBatteryStaple472 10 ай бұрын
I love the grin at <a href="#" class="seekto" data-time="363">6:03</a> with the inverse hyperbolic tangent. Math is fun and your subtle humour like this makes it fun.
@mrizwan4828 9 ай бұрын
😅
@ahmedalhomaide4416 10 ай бұрын
Bro, I just graduated from high school but I can't ignore your video titles... Great as USUAL!
@blackpenredpen 10 ай бұрын
Thanks!!
@ethancasillas7325 9 ай бұрын
There is absolutely no reason for me to have watched this, I'm not in -or ever going to take- any calculus class where I have to integrate the quadratic formula. Yet here I am, loving every second of this video :)
@blackpenredpen 9 ай бұрын
Thank you!
@skyethebi 9 ай бұрын
<a href="#" class="seekto" data-time="328">5:28</a> You can also use partial fractions for this integration u^2 - b^2 = (u - b)(u + b) 1/(u - b)(u + b) = 1/2b(u - b) - 1/2b(u + b) And that way you don’t end up with inverse hyperbolic trig
@xninja2369 9 ай бұрын
I was gonna write sameee
@perfectman3077 9 ай бұрын
Nerd
@xninja2369 9 ай бұрын
@@perfectman3077 stupid Sigma 10 year old kid , your place is not here go somewhere else dude 🤡 If you want to learn something than learn, if don't than get out ..
@joyboy69lffy 9 ай бұрын
so u will get 1=A(u-b) +B(u+b) so you cant find the value of A and B since you dont know neither the value of u nor b
@skyethebi 9 ай бұрын
@@joyboy69lffy My bad. You can still use partial fractions since b is constant with respect to the integration but you need to divide by b. I updated my answer so it should be right now.
@acuriousmind6217 10 ай бұрын
A much more interesting approach is differentiating each quadratic formula with respect to each variable (a, b, c) and observing how each variable contributes to the fluctuation and the identification of the sweet spot for every variable that results in a more diverse set of solutions.
@dekippiesip 9 ай бұрын
Sure, a lot of cool things could be done with this. Take the solution set as a function with 3 variables, include complex numbers if you like, and look for all sorts of patterns. I'm thinking of stuff like maximizing the distance between 2 solutions for all vectors (a,b,c) of fixed length and shit like that. It's a heavily unexplored topic!
@jamesdcarroll1 9 ай бұрын
Its been 30 years since I took this in college and I can still fall asleep halfway through.
@Alkis05 9 ай бұрын
Who said you would forget how to do the things you learned to do in your freshman calculus course?
@MathNerd1729 10 ай бұрын
The last part reminded me of when I asked 2 different AI models (not ChatGPT) about whether they can prove Euler's Sum of Powers Conjecture . . . both of them had factual inaccuracies in their responses and one of them even claimed that it's currently unknown
@acuriousmind6217 10 ай бұрын
its the same as asking google a long question... at the end of the day its trained on pattern recognition you have to de-personify the ai so to speak when approaching it with complicated questions
@decaydjk8922 10 ай бұрын
All they are is token prediction systems, just a more advanced version of the predictive text in your phone. They can't "do" math or anything so no, that's not surprising
@abuabdullaahiwaaaishatah8235 9 ай бұрын
Lol
@lolerie 9 ай бұрын
Because only GPT 4 is AGI. Above human level.
@sunnohh 9 ай бұрын
Yeah ai is dogshit at medium hard or slightly complex
@gamingbeast5755 9 ай бұрын
Math Professors are the real JOD people on Earth....
@abyzdoof8821 10 ай бұрын
i love how we can just see all of those dry erase markers on standby in the bottom right. video’s fantastic as always, keep up the great work!
@dayzimlich 9 ай бұрын
Great stuff, bprp! I love your videos. You teach but you do not talk down to the audience and you bring a catchy sense of humor and fun. Here's to your continued success!
@blackpenredpen 9 ай бұрын
Thank you very much for your nice comment!
@TranquilSeaOfMath 10 ай бұрын
Love the video. Easy to follow; well explained. I also like how you show the technology failure at the end.
@gachanimestudios8348 10 ай бұрын
Legit the first question i asked from that title was "with respect to what?"
@DatBoi_TheGudBIAS 9 ай бұрын
ive figured out a long time (by curiosity) dat chatgpt absolutely sucks with math problems. it can fail in something as simple as a quadratic equation, let alone a full integral
@MayurAvad 10 ай бұрын
Really enjoyed this one !
@blackpenredpen 10 ай бұрын
Glad to hear. Thanks!!
@tuseroni6085 9 ай бұрын
that trick of adding in a -b and a +b blew my mind. it makes perfect sense in retrospect, -b+b=0 so if you have something that is also something + 0 and thus you can put anything in so long as it cancels out to 0...
@blackpenredpen 9 ай бұрын
Yes 😃
@winghei10 9 ай бұрын
Teacher: Maybe there is a mistake. Please verify the answer by differentiate the answer.
@jok2000 10 ай бұрын
If you want ChatGPT to do advanced stuff you tell it to write a python program that does what you want like plot a function, do a symbolic integration or even get it to create a rotating globe in OpenGL. I've done all of these things with it and more... a subway station mapper using google maps API and TKinter etc. Some, like the maps API require a bit if tweaking but not the integrator.
@zelda12346 10 ай бұрын
"I integrated the quadratic formula" has the same energy as "I sawed this boat in half!". And since you differentiated this already, "I sawed ANOTHER boat in half!"
@pulkitgupta5367 10 ай бұрын
Its giving the same energy as that flex tape ad where he cuts a boat in half and then tapes it back together: kzfaq.info/get/bejne/Zt6qgZl3spvcj3k.html
@seanoneill2098 10 ай бұрын
You are doing good things !!
@panhandlejake6200 9 ай бұрын
So definitely an academically interesting exercise. Is there any insight about quadratics and their solutions that can be learned from this result?
@dhruvaraju5870 9 ай бұрын
No!
@A-0II0Io 9 ай бұрын
😂😂😂 the last part of your amazing video was amazing. Wow...
@granthqweqw5244 9 ай бұрын
Would love to see triple integration with respect to all variables there!
@mynamejeff69302 9 ай бұрын
Last part of integration with u^2-b^2 can be done with writing it as( u+b)(u-b) and then matching the numerator to get 1/2b^2 ln(u-b)/(u+b)
@SomeoneCommenting 9 ай бұрын
It would be interesting to compare the integrals of da, db, and dc for the same curve parameters to see what happens.
@anonymousaccount-27 10 ай бұрын
Its a fact that ChatGPT has cracked the toughest exams but does dumbest mistakes in new type of questions....
@elquesohombre9931 10 ай бұрын
Those exams aren’t usually logical ones, that and it will have many resources that allow it to “cheat” on those exams by pulling data from sites that have very identical questions/answers
@jok2000 10 ай бұрын
Just ask it to write a python program to symbolically integrate.
@SlipperyTeeth 10 ай бұрын
Me holding a stack of answers to every exam ever published online: It's a fact that I can answer the toughest exam questions, but am unable to apply any reasoning skills to "new types of questions".
@brookek3116 10 ай бұрын
@@elquesohombre9931Yep, I’d do pretty good on a lot of exams too if I was able to copy and paste answers from the internet
@Craznar 9 ай бұрын
Given it is only trained on language, that it even attempts to work out the answer is impressive.
@perspicacity89 9 ай бұрын
Fantastic video, thank you.
@thbb1 10 ай бұрын
Beautiful, but can we find a geometric interpretation of this integral? Or at least use it to gain some insights on the regions where the determinant is positive or negative as a function of the first coefficient of the trinome?
@gileadedetogni9054 10 ай бұрын
Maybe f(a)=all the formula
@Jordan-zk2wd 9 ай бұрын
Someone correct me if I'm wrong, but utilizing the intuition of the mean value theorem I think this means: if you vary a from some value a0 to a1, then the mean value of x which solves ax^2+bx+c=0 will be [f(a1)-f(a0)]/[a1-a0], where f(a)=-(b/2)ln|a|+sqrt(b^2-4ac)-(1/b)tanh^-1(sqrt((b^2-4ac)/b)* . I tried running it on desmos, I'm tired so I might have did this wrong, but here's an example of what I mean: from a=1 to a=2, the mean value of one of the roots of ax^2+3x+5=0 is -2.66324ish. I did have to throw in a negative sign at the end to make it all work, which means I'm made some small error along the way, but visual check of a few examples seem to confirm that it does work. * Important caveat, this will be the average value of one of the roots as you very a, the other one would have a slightly different definition of f(x) for the choice of the negative root of b^2-4ac, and thus give you a different average value corresponding to that root. @blackpenredpen ?
@gamingbeast5755 9 ай бұрын
Nice Explanation Sir...
@OleJoe 10 ай бұрын
This is wild, man!
@ivanrodionov9724 10 ай бұрын
Is there a physical interpretation to integrating the quadratic formula with respect to the polynomial? Does it tell us anything special?
@spearmintlatios9047 10 ай бұрын
If we were to create a graph with an a axis and an x axis, where x is vertical (where y usually is) and a is horizontal (where x usually is), this would tell us the area under the curve of the graph. x = (-b+sqrt(b^2-4ac))/2a in terms of b and c which are constants. That’s really it, it’s not too special. If you wanted to see what this graph would look like you could go on Desmos and substitute x for y and a for x, and then create sliders for B and C.
@ivanrodionov9724 9 ай бұрын
@@spearmintlatios9047 Well that it can be viewed as an an area under a curve is one interpretation, however this is not what I am asking. He is doing something much more fun here, see, the quadratic formula is a function in polynomial of 2 degree space which has 3 degrees of freedom, a,b,c. It therefore maps a unique polynomial represented by the point (a,b,c) in the space and maps it onto two values which are it's roots. Now, when he integrates the quadratic formula, with respect to a, this is the question on how it changes the mapping and what does it tell us now. I am 99% sure there is quite a bit more going on here.
@spearmintlatios9047 9 ай бұрын
@@ivanrodionov9724I kinda get what you’re saying but I’m rusty on calc 3 stuff tbh. If you want, you can represent the quadratic equation as a graph mapping the function of a, b, and c to a value of x. or f(a, b, c)=x. this would be a 4 dimensional graph which is somewhat beyond our imagination, but everything should still apply. Since we only integrated with respect to A here, B and C are still able to be changed within the integral. pretend that B is equal to 1 for now. Then we can have a 3D graph of f(a, c) = x. In this case, we have a the a and c axis representing our domain, and an x axis representing our range. So for any point on the (a, c) plane, the output of this function is some “height” in the x axis creating the point (a, c, x). If this is the case, and we assume B is constant, then this integral represents the area summation of the height from the x output to the flat ac plane from a given start and end point a0 and a1, So it would still just be a 2D area found from an infinitesimal slice of a 3D graph. It is cool that we could move this slice by changing our bounds of a0, a1, and C. You could imagine changing these bounds as moving and stretching a tall piece of paper through the 3D space of the function, and calculating the area of that paper if you slice off all negative values and everything above the x points. To calculate the volume of the shape this graph makes, we would have to integrate by another bounds as well, a double integral. But the same logic works as well when we include B again. The integral would simply be a slice in a 4D shape that has X as the output. Unfortunately, we can’t just set X to 0 and have a graphical interpretation of 0 = integral (g(a, b, c))da. I don’t think there actually any representation there? If we wanted to find an integral that works here we would probably have to move some terms around before hand and treat it as a differential equation. Correct me if I’m wrong
@spearmintlatios9047 9 ай бұрын
@@ivanrodionov9724 after reading what you’re saying again I see you’re talking about something entirely different. Is there any tool that could create a mapping similar to what you’re talking about? I think what I was discussing might relate when you consider he only integrated using a plus and not a minus
@minseok8726 9 ай бұрын
@@spearmintlatios9047any applications in Engineering could you think of? Aerodynamics maybe
@99chartered 9 ай бұрын
Thanks so much, it was fun!
@disgracedmilo 10 ай бұрын
the sequel we needed
@muralidharrangaswamy9643 9 ай бұрын
I think you can also expand the denominator of the last integral into partial fractions and then express the result as a difference of two logarithms
@madbrad6282 9 ай бұрын
That was fun to watch.
@guliyevshahriyar 9 ай бұрын
Very good lesson, thanks.
@Tasz_ 9 ай бұрын
Came for blackpenredpen got blackpenbluepenredpen what a day to be alive
@arsalmathacademy 9 ай бұрын
I am always fan of this great person, brilliant brain
@arsalmathacademy 9 ай бұрын
@@English_shahriar1 Thanks
@Krish-su4oh 10 ай бұрын
Last part was very meaningful 😂😂❤
@NonTwinBrothers 10 ай бұрын
Im glad I woke up at 3am to enjoy this 😂
@alvaroarizacaro3451 9 ай бұрын
Muchas gracias. Muy bonita esta integral.
@Protactinium91 10 ай бұрын
I’m looking forward to learning hyperbolic tangent!
@merixan9322 10 ай бұрын
I literally just watched the differentiating video of the quadratic formula yesterday
@holyshit922 10 ай бұрын
<a href="#" class="seekto" data-time="369">6:09</a> If u is in interval (-1,1) but if u is outside this interval you have b/u as argument of inverse hyperbolic tangent
@inverted2533 6 ай бұрын
you know what, im startin to like this guy
@adrienanderson7439 10 ай бұрын
Another one that's fun is the limit as a goes to 0, It gives an interesting result that makes sense when you think about it. If you really want a challenge you could do the limit of the cubic formula as the x^3 coefficient approaches 0 to see if you can get the quadratic formula
@euler1 10 ай бұрын
I did that like 2 month ago and thought it was very nice how the limit works out at the end. I don't know what you mean with the coefficient of x^3, what equation will you apply the limit to?
@adrienanderson7439 10 ай бұрын
@@euler1 there is a cubic formula but it is kind if complicated
@adrienanderson7439 10 ай бұрын
@@euler1 The solutions to ax^3+bx^2+cx+d=0 is given by the following: let u=b^2-3ac, let v=2b^3-9abc+27a^2d, let w=((v+-(v^2-4u^3)^(1/2))/2)^(1/3) where we can use any root for each cube or square root. Then x=(-1/(3a))(b+w+u/w) gives the roots
@euler1 10 ай бұрын
@@adrienanderson7439 It's the first I see this thanks for the explanation, I am curious to try it out but I think I will just look up a proof for it as I don't have that much time xd
@adrienanderson7439 10 ай бұрын
@@euler1 Yeah honestly I haven't done it myself and I tried to today but it really gets into the weeds
@cameronskellams5670 10 ай бұрын
Thats hectic! Love the definitive proof at thr very end that chatgpt clearly has discalculia 😂
@jok2000 10 ай бұрын
Just ask it to write a symbolic integrator in Python.
@eliteteamkiller319 5 ай бұрын
The ending had my rolling.
@xjk.- 10 ай бұрын
can we multiply both the num and denm with sqrt(b² - 4ac) when we split the integral -b/2a da +... and see what will happen?
@michaelbaum6796 10 ай бұрын
Nice solution - great👍
@blackpenredpen 10 ай бұрын
Thank you!
@justanotherguy469 9 ай бұрын
I love your t-shirt.
@dr0g_Oakblood 9 ай бұрын
Last segment is a perfect example of ChatGPT just making stuff up lmao
@Jesuisunknown 10 ай бұрын
I don't understand a lot but I'm getting interested again to learn the basics of Calculus
@Noobthepro0 9 ай бұрын
Instead of using inverse hyperbolic function as the integral of 1\(u^2-b^2), you may also use ln{(u-b)/(u+b)}/2b. That's how GPT might have also got the answer in terms of natural log
@alijhi 10 ай бұрын
Brilliant !
@fredrickfred4621 5 ай бұрын
You can also use trig sub to avoid hyperbolic tan
@General12th 10 ай бұрын
Hi Dr. Pen! Very cool!
@blackpenredpen 10 ай бұрын
thanks!
@francaishaitam6708 3 ай бұрын
for the inverse hyperbolic tan part,you've better to go with a partial fraction decoposition ,since we don't now the domain of integration , it's ok to go with the inverse hyperbolic tan only if a is from (-1,1)
@parzflash1619 9 ай бұрын
We got the integral but can you explain the solution graphically as integration with limits gives area and what would the area of this graph be
@GuilhermeSilva-or8ud 5 ай бұрын
amo suas questões 🫶 um abraço do Brasil
@vaccino3359 8 ай бұрын
Applied maths: Useful things that would probably come handy in life Pure maths:
@yigiteldek 10 ай бұрын
i retried asking the integral to gpt couple times and with some hand it eventually integrated the formual exactly as in the video eventually
@Saber09 5 ай бұрын
I am drunk right now but dude this video is absurd, it would never get to my mind this solution! Great! Have a good day :D
@said_qurbanov 9 ай бұрын
Bro will integrate my life
@the_physics_pro.sohamkakka7765 10 ай бұрын
Wouldn't it be better to do trigonometric substitution in the 2nd integral instead of substituting the root as an algebraic variable?
@romanbykov5922 10 ай бұрын
very interesting video, blackpenredpenbluepen! :)
@blackpenredpen 10 ай бұрын
Thank you!!
@tomasbeltran04050 10 ай бұрын
It's been a while since I last saw one of your videos
@promethius7820 9 ай бұрын
This integral is basically the unbounded interval is basically the terms for the "missle knows where it is" problem, isnt it?
@deldc 9 ай бұрын
My eyes are burning! Too many knowlegde! This is insane!!!!
@modolief 9 ай бұрын
Very charming, thanks 😃😃😃😃😃😃😃
@tatoon34 9 ай бұрын
Basically integration is getting the area under the curve or easier formula to understand and so what is the area under the curve graphically or does it make it easier to realize after doing integration?
@user-ek9vo2ub9b 9 ай бұрын
So what can this be applied to specifically? Curious.
@benberlowitz6381 9 ай бұрын
Please do a video where you triple integrate with respect to all 3 im genuinely intrigued what the result would be (is it even possible??)
@jannegrey593 5 ай бұрын
Have you ever tried differentiating/integrating quadratic formula? Yes, basically as soon as I learned how to do basic derivatives from limits and noticing that circle/sphere calculations look like they have been integrated/differentiated using "power rule" (I think that's what it's called). 2*pi*r, pi*r^2 and 4/3 pi r^3 looked too much like coincidence. So I looked at quadratic formula (I also wondered back then what would happen if I did look for zeros when "delta" is "negative" - of course without imaginary numbers it is "harder" to do. Still doable in some cases and if you don't mind going a bit outside real numbers for rules). Sadly my knowledge was insufficient - had I tried it when I was more proficient it would likely help.
@adarsh5997 9 ай бұрын
<a href="#" class="seekto" data-time="339">5:39</a> but the integrals of the form dx/(x²-a²)= 1/2a*ln(x-a)/(x+a) where ln is natural log
@danielsaldivar5622 9 ай бұрын
Doing math with a smile!
@fernandofernandez-sv7tr 9 ай бұрын
Excellent
@kemico1272 9 ай бұрын
awesome!
@Alkis05 9 ай бұрын
Wolframalpha still has a lock on the freshmen calculus student market.
@composerlmythomorphic2635 10 ай бұрын
I got another (perhaps complicated solution: Suppose the integrand is x, where y=ax^2+bx+c. Then dy/dx=2ax+b, and dy/da=x^2. Hence da/dx=(2ax+b)/x^2. Then integral=int[(x)(2ax+b)/x^2]=int[2a+b/x]=2ax+b lnx +C
@cheeseparis1 10 ай бұрын
Thanks, this is what I needed to prove the last Fermat theorem. This may not fit in a KZfaq comment actually.
@patrickford7582 9 ай бұрын
I have a question, for u^2 -b^2 what if you replaced the constant b with say ib so that (ib)^2 was -b^2, then substituted it back it? That is use u^2 + (ib)^2 for u^2 -b^2.
@maazqayyum693 9 ай бұрын
Could we not jist substitute u square with sec in u squared minus b squared?
@AdakGamer 9 ай бұрын
Sir we can instead use 1/2u . ln[x-u]/[x+u] + c for the integration of 1/u^2 -b^2 du
@AdaDenali 9 ай бұрын
I am struggling to interpret what the area under this curve would represent.
@PunyaMalhotra 9 ай бұрын
The last integral dU÷U²-b² Is equal to (1÷2b)×ln|(u-b)÷(u+b)| The above result is definitely okay, idk about the one you used(i am nkt saying yours one is incorrect, its just that i never saw that formula😅)
@phoenixspirit6014 10 ай бұрын
Really nice
@MathFromAlphaToOmega 10 ай бұрын
Cubic formula next, please!
@blackpenredpen 10 ай бұрын
😆
@berenedain8427 10 ай бұрын
"He's too dangerous to be kept alive!"
@teddy05p 10 ай бұрын
Great video !! I have an interesting problem for you:) Infinite series from k =2 of 1/(ln(k)^(ln(ln(k))) we are only allowed to use limit comparison or comparison. :)
@tomctutor 10 ай бұрын
lim (k=2 .. inf) f(k) = A + lim(n=27..inf) f(n) where A is some number and f(n) = 1/(ln(n)^(ln(ln(n))). Each term in the expansion of f(n) is now strictly positive, since ln(ln(27)) > ln(ln(e^3)) = ln(3)>ln(e) = 1, i.e. positive. So f(n) series terms are bounded above by 1, so only need to consider a monitonic decreasing series. Oh and also determine the leading constant A value.
@teddy05p 10 ай бұрын
@@tomctutor numerically this series diverges, so it is not bounded by 1.
@teddy05p 10 ай бұрын
@@tomctutor oh sorry misinterpreted your statement, it is, youre talking about the terms :)
@tomctutor 10 ай бұрын
@@teddy05p Yes I did not comment on wether or not the series converges. If alternating then it would converge, but it isn't alternating as I shown.
@jacobblum618 9 ай бұрын
Could you do double integration to integrate based of two variables?
@blackpenredpen 9 ай бұрын
You can see my 100 integrals part 2. I did about 20 double integrals there toward the end.
@n4p3r0 10 ай бұрын
You should do it with respect to b and c next :3
@mqb3gofjzkko7nzx38 10 ай бұрын
I just noticed the hundreds of black and red markers in the bottom right corner.
@blackpenredpen 10 ай бұрын
😆
@arnold-pdev 9 ай бұрын
Worth noting this function has a domain restricted to the interval 0 =< 4ac =< b^2. The second inequality comes from the discrimant (existence of roots). The first is a statement that either a and c are both negative or both positive. This is weird to me!
@sless6928 9 ай бұрын
I'm just amazed you found 3 working whiteboard markers.
@chakonleung8675 7 ай бұрын
What is the implication for this integration? Any significant findings about this?
@DiggOlive 10 ай бұрын
What does this look like if you plot it in 3D?
@pradhyumnajadhav9137 9 ай бұрын
please do triple integration on this next
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