Disc/Washer Method vs. Shell Method (rotated about different lines)
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- Опубліковано 28 вер 2024
- Volume of Solid of Revolution rotated about different lines. Disc method vs. shell method for calculus 1 or AP calculus students.
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Hope this helps with your calculus 1 or AP calc
Time stamps:
a) area of the region, both vertical and horizontal rectangles 0:06
b) rotate about x-axis, 5:58
c) rotate about y-axis,15:16
d) rotate about x=5, 21:51
e) rotate about y=3, 30:56
This is not a difficult questions...I know better then you
@@al-quran9845 No, it doesn't work that way... stop saying this that's nonsense.
Mathematician Way Don't be an asshole. This isn't about whether it's easy or not. It's a video meant to help calculus students who are struggling, it's obviously not meant for people such as yourself.
CAN U DO WITH X=3 ALSO??
Where's the polar rotation ?
This man may have taken the GOAT throne. No shade to The OChem Tutor or Sal Khan (they're both great and have done a lot for me), but NO ONE has been able to teach me washer method until this man. Thank you Blackpenredpen.
ochem did
Theyre goats
You sign up to a college only to learn it from UA-cam. It makes sense. Thank you.
I’ll be doing the same
Wait you learn this in college isn't this a high school course?
@@78anurag this subject is generally learned at either the end of Calc 1 or beginning of Calc 2, or both, but this just depends on what calc 1 and 2 course you took. But people still take calc 1 in college, so it would make sense either way to be learning this in college
@@jamesarnold3286 I learnt this in Calc 1
fuck me too and im doing that 2am :)
Great video! I love all the UA-camrs making math more easily understood. Just a point of clarification... "Washer" refers to a fastener used in combination with a bolt and nut, not a washing machine.
Omg, that washer?!
I always thought that as the washing machine washer lol
A great help visualizing when online classes don't do nothing!
You made this so easy to understand. Up until this point I did not really understand these methods at all but now I actually understand what I'm doing. Thanks!
You are a king! You explained chapter 7.2 of the Stewart Calculus textbook in 30 mins. Thank you sir!
this has been a great help to actually understand the concept, i am very thankful for this video my good sir
love u broooo, youre strongest math teacher Ive ever seen
Wonderful!! I was struggling to understand the topic
But you make it easy, I tackle questions from any angle
Your explanation is Amazing .
Thank you. I am glad to help.
I blanked out for 2 hours trying to learn this in class and fully understood you in less time. Thank you so much!
this video helped me a lot today was my class and hardly understand anything you helped me thanks!!
One of his best videos for sure!
Excellent video...
Perfect explanation...
Love from India 🇮🇳 ❤️
YOU JUST SAVED ME! ALL HAIL!
This video is incredibly good.
YOURE THE BESTTTT THANK YOUUUU FINALLY SOMEBODY DID THIS RIGHT
Next time please show us how to find the volume of rotation/revolution about a SLANTED LINE (like y = x/2 in this case).
Rotate the function
@@johnny3475
Right. Show us how.
How about rotating around the line y=5-2x ?
(i.e. perpendicularly intersecting y=x/2 at (2,1) )
thank you for this, i really couldn't get it when only reading the module
Math exam tomorrow thank you so much bro 🙏
this is so much more clearer than what my teacher had taught me. thank you!
Great video!! Helped me figure out how to set up my equations
Glad it helped!
Very good presentation
I understood very well
You're excellent illustrator (Y) ❤️❤️❤️❤️❤️❤️❤️❤️
I did a volume of solid revolution around the line y = -x just to crank things up a notch. It took a while since I had to rotate the region clockwise π/4, then do a revolution around the y-axis with shells. It's a bit of work, but doable. Unless I made a mistake somewhere I got 52/15 * √(2) * π.
I love this guy.
I needed to review this topic and my goal for this week is to solve 100+ integral problems 😁 nice timing!
thank you for explaining!
saved my calculus grade once again
I might be early for the video, but you are late... 2 months late, when this topic was presentend to me for the first time. As always, excelent video.
Thank you♥️
THANK YOU! I was about to bomb a Calc exam tomorrow. Whew
Best luck to you!
Please create the video L hospital rules maths
Thank you! it's really helpful!!!
Small correction for you.......the "washer" method is so called because of a washer....a small flat annulus used with a nut and bolt to spread the load over a greater area, not a washing machine
Absolutely legendary
This is great we just did this in bc calc class
Man you saved me from failling thanks a lot... i still dont know if i failed or not but i finished the exam 😂
Thanks so much for this
Thanks man! huge help
I love you so much
"washer method, because, just like washer machine, hole in middle",,,,,LOL! classic!!
very nice job
you are the best.......
Just a little bit of confusion here sir. Shouldn't we also subtract the volume of the the cylinder at the center with the diameter of 2 from x=4 to 6 with a height of 2. Because as always, I answer your problems first to test my intuition and thinking and based on my initial take, at the end I subtracted 2π cubic unit as the vol of the cylindrical hole. My apology If I had mistaken the concept and thank youuu
You're always great Mr. Steve. A friend of mine from the Philippines said that you gave him a shirt for noticing the error and correcting on one of your inverse laplace equation. Just for clarification, this is not my attempt on getting a shirt from you, this is just clearly my confusion on the concept hahahaha
Great video, thanks :)
At minute 30:56, you integrate from 0-4. Why is it 0-4 and not 0-2?
Hello sir
Hope i found you well. I have a question for you to kindly assist.
When f(x) is divided buy (x+3) the remainder is -6 and when the same function is divided by (2x+1) the remainder is -9. What is the remainder when f(x) is divided by (x+3)(2x+1)?
Zimsec questions in Zimbabwe
Thank you.
-6/5(x+8)
This isn't an extremely important question since I understand how to set it up and maybe it is just coincidence with this example but the shell method on the x-axis, I set the integral up wrong at first and got the same answer. I did (2y-y^2)dy.. and then multiple by 2pi after .. instead of (y (2y-y^2))dy . Just wondering why that worked when I missed the height component. Then when rotating about the y-axis.. I get slightly different numbers for the washer vs shell method. Only (.00000011) difference but it is bugging me and I want to know how it could be off. I do know that the calculator is just estimating so maybe that the reason.
amazing video
how would you find the volume of revolution of 2y^2 -6rootx + 3 =0 the curve is above and below the x-axis if rotated around x axis. Do you multiply by pi/2 since you only need to rotate it pi rad to complete a solid? From x=0.25 to x=4
Hey sir u r from which country !!
I also love maths n physics !!
U r videos are awesome !!
can you rotate around axis that are not constants such as y=1/2x or other functions, it would be a little weird trying to rotate around a curve like y=x^2
When do we use disc method and when do we use shell method?
He used both Shell and Disc method to solve the same problems. Which so, you can use any method thats easier for you!
For me, Washers method is more basic and easy.
I’m still a little confused on when I should use dy and when I should use dx? I guess I’m not sure how to figure out what world I’m in lol I’m terms of dx and dy
Why don't you make videos on geometry.
By the way, that's a nice Luxury poke ball you have there.
30:29
I literally just finished Calc II a few weeks ago lmao
Lmao we literally just went over this in calc BC
He really means washer as a washing machine? All this years I had been thinking the shape looks like a dang washer for a bolt. :)
His explanations are on point, tho. Thank you!!!
If this guy didn’t exists, I would fail all of my calculus courses.
haha
free Israel
I love how passionate he is when he’s doing maths
Ik it’s making me happy how happy he is
This is the only person who’s even come close to being able to explain shell method in an understandable way
A washer is a thin plate with a hole (typically in the middle) that is normally used to distribute the load of a threaded fastener, such as a bolt or nut.
en.wikipedia.org/wiki/Washer_(hardware)
Panos Triantaphillou omg, thank you! I always thought it as the washing machine washer lol.
Yes, _"Washer method"_ is not after someone's name. But we may suppose it is *not* related to _"washing machine"_ as in _"washer"_ , a person or device that washes something, and as in _"wash"_ verb, clean with water and, typically, soap or detergent.
It is more related to _"washer"_ , a *flat ring* fixed under a nut or the head of a bolt to spread the pressure when tightened or between two joining surfaces, since we subtract a smaller disc from a bigger disc, we got a *washer disc* or flat ring. Do you gents agree?
@@alexdemoura9972 It makes a lot of sense, must be because of this.
I actually laughed out loud when he said washing machine, i thought he was joking lol
May I ask.. Is circular ring method and washer the same?..
I cannot express enough my gratitude for your videos! I struggled to conceptualize washers/disks and shells, but with your thorough explanations, I could tackle these questions with ease. Thank you!
I am glad to hear. Cheers!!
Yes, thank you, we skipped all of this because of the pandemic in my calc class
this man just taught me three weeks' worth of class in under an hour. That's crazy
3 weeks? my university taught us this in less than a week and expect us to be able to understand it lol. My university's curriculum is insane. (Not gonna expose my university, but it's 1 of the top 5 universities in asia as of 2024.)
Hands down best video about this on UA-cam.
Subbed! :)
What do you guys think of this? Here is how I understand it:
VOLUME FORMULAS FOR WHEN IN TERMS OF X
AREA: ₐ∫ᵇ h dx
ₐ∫ᵇ[f(x)-g(x)]dx
WASHER: ₐ∫ᵇ π(R - r)² dx
ₐ∫ᵇ π[x₁-f(x)]² dx - ₐ∫ᵇ π[x₁-g(x)]² dx
SHELL: ₐ∫ᵇ 2πrh dx
ₐ∫ᵇ 2π [x₁-x][f(x)-g(x)] dx
f(x) = outer function
g(x) = inner function
x₁ = x-coordinate at the axis of rotation
Learned this like 2 years ago and watched the whole thing anyways for a refresher lol good video
Haha same. I didn't get a full grasp of learning this but it was really fun. Everything in previous math used for calculus. Can't believe I thought it was generally difficult.
Ever since The Organic Chemistry Tutor started paywalling his videos, I've been struggling in math. Thank you for this.
i thought i would never find someone like him, but guess what i just found the one BLACKPENREDPEN
I can't believe that a few years ago I watched your channel, and I had no clue what all those hieroglyphs were on the whiteboard, And now I'm watching your videos for an explanation! keep up the amazing work, I love your videos! thanks!
I really appreciate that you dive deep into simple questions to ensure the fundamental understanding of the topic.
God Bless his soul. He just saved me for my test tomorrow. Thank you for your service sir.
this man draws better than my professor at Purdue
Hahha. I will come back and verify this comment after next fall😎👏
man i'm brazilian and my classes were suspended due to the covid-19 pandemic, your classes have helped me a lot, here we see this in calculus 2 continue with this incredible work and may god bless you
kkkkkkk br realmente ta em todo lugar
Acompanho esse canal desde que comecei a estudar matemática direito hue
Isso aí é cálculo 2?
@@andrewcarvalho9158 sim mano, calculo de volume de solidos usando integral
@@andrewcarvalho9158 Eu vi em cálculo 1 mesmo
Hey Can U Do With X=3 Also Pls
This video needs to be the first thing lectures show to their students 🤣🤣🤣Very much descriptive and straight to the point, it even helps train the mind how to visualize the solids of revolution. Keep up the good work 🙏🙏🙏
"Just like washing machine, Washer has hole in the middle." - BlackPenRedPen - 2020
i've never commented on a youtube video before, but dude i was depressed at my final exam week and you made my day with your smile. thank you! keep doing it. you are amazing at your job.
you can really see how passionate he is for teaching math through this video. great work, keep it up!!
I appreciate that!
I think i have an interesting question.... How would you do this problem if you had to rotate y = sqrtx and y = x/2 about ******x=3******...i cant really seem to figure it out..
I see the cause of your 'problem': after 180 degrees of rotation, the region envelops part of its original self.
I suspect the way to 'solve' this is by not double counting the regions, but do count them at least once, i.e. only look at the boundaries (one of which can be r=0 x=3, because the radius can't be negative) to derive the formulae for your integrals.
By the way, do you think you can find the volume of that same region between y=sqrt(x) and y=x/2 once rotated around the line described by y=10-2x ?
edit: I just tried your problem via the disc/washer method and got my formula's but then also noticed that when I'd tried to do it via the 'shell' method, I would not be able to find a single formula for the shell length a.o. due to the shell being split in two pieces for 2 < x < 8-2sqrt(7)
edit 2: Here's my results & how I got there (I shortened *sqrt()* to *rt()* and _integral of f(z) from z=a to z=b_ to *igl(a,b, f(z) )* )
Preparations:1) what does it look like before rotation:
- bottom boundary is line A described by y=x/2 x=2y
- top boundary is curve B described by y=rt(x) x=yy
- A and B intersect at (0,0) and (4,2)
2) what does it look like mirrored vs. x=3 after 180 degrees rotation around x=3:
- bottom boundary is line C described by y=(6-x)/2=3-x/2 x=6-2y
- top boundary is curve D described by y=rt(6-x) x=6-yy
- C and D intersect at (6,0) and (2,2)
3) how do these interact?
- for x=3 A and C and B and D are equal with the value y=1.5 and value y=rt(3) respectively.
- C intersects with B left of the rotation axis at point P with coordinates (8-2rt(7),rt(7)-1) = (xp, yp) (by definition)
Resultant formula:
1) Disc/washer method, in this case becomes two wash-machines standing back to back:
V = igl(0,yp, pi(3-yy)^2 ) + igl(yp,2, pi(2y-3)^2 ) - igl(0,3/2, pi(3-2y)^2 ) - igl(rt(3),2, pi(yy-3)^2 )
(i.e. outer boundaries described by B and C minus inner boundaries described by A, D and r=0 x=3)
2) Shell method, subdivided in different divisions having different formulae describing the (total) lengths of the shells:
V = igl(0,2, 2pi(3-x)(rt(x)-x/2) ) + igl(2,xp, 2pi(3-x)(rt(6-x)-3+x/2+rt(x)-x/2) ) + igl(xp,3, 2pi(3-x)(rt(6-x)-x/2) )
(That is B-A, D-C+B-A and D-A respectively.)
Apollorion holy crap thanks for answering with such detail🙏🏻
@@anirvinvaddiyar7671 To me it was mostly just time spending during a lock-down, and I hope some one will also check whether I did not make a mistake.
Did you understand? And if so, did you find no mistake?
I want to send u a problem of integration..how can i send u???btw i love ur vedio..(ur suscriber from India🇮🇳
Sorry if I misheard this, but you were talking of the "washing-up" kind of "washer" at one point. Just in case that's what you're trying to relate the method name to, here's Wikipedia on the nut, bolt, and washer, versionr, which fits the usage here better. en.wikipedia.org/wiki/Washer_(hardware)
I think this interpretation of the "washer" in 'washer method' would be well correct if the two integrals were fused into one, but not really when kept separate.
You deserve the Presidential Medal of Freedom.
Math is an integral part of my life,is is for yours?
This video being uploaded in my birthday is one of the best things that happened to me today, so yes.
Math am part of my life
math has been a great addition to my life
Yeah my life is the derivative of maths too
Maths is an interal part of my life and differentiates me from others.
谢谢你救我狗命 感谢你 thank u very much
Este video debería llamarse “lo que tu profesor de cálculo 2 no te enseña”... lamentable .... Gracias maestro BPRP💪
I am just getting interest on the subject, just by seeing him...what ah positivity his face has!!!! Wow!!!
Thank you for your time. If it wasn't you I would have not understood all these. *Highly appreciated from South Africa*
Nowadays u r using blue pen😑😑
thank you dude alot you saved my ass alot hahahahaha
Exam in like 2 hours and I swear I have watched this video along with a few other of his like at least 3 times each and It has really helped me get the logic down. Fingers crossed I can apply the logic I learned on the exam
Best of luck!!
Am I the only one who finds this extremely tedious? xd. Great video, so helpful bprp, love from Spain!
Thank you. It really helped me
I'm not in Calculus yet, but I love watching your videos regardless.
BrownWater ayyyy 🤟me2
the xr comment LMAO thanks so much for this video, ur so engaging and I’m starting to understand this better now!!!
I have a test about this today. The video recomendation was brilliant i think im ready now for it. Thank you Blackpenredpen.
Thank you for using the same two functions for the different examples, I hate when some videos try to explain the different rotations but pick a bunch of different functions for each method and rotation, this makes it crystal clear to understand the fundamental concepts without having to rewatch or pause a lot.
Make videos on vector algebra please!!! Or Vector-3D
you are a frkn rockstar thank you so much