Calculating Volume by Cylindrical Shells
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- Опубликовано: 11 сен 2024
- We now know one method for finding the volume of a solid of revolution. But there are tricky examples where the normal method won't work, like when both the inner and outer radius of a washer are being determined by the same function. Luckily there is another method we can use! It involves cylindrical shells. What's that, you ask? Watch this!
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Your 7 minute explanation with visuals helped me understand way more than the textbook's long text and the teacher's 1 hour lecture.
CP SLO
I couldn't imagine what was happening for the life of me... this made it so clear, thanks a lot man.
This is probably the 3rd video that I am watching and I think all the animations that you use are very beneficial! Such videos are very helpful and they cover the basic of that topic in a very less time and that is highly appreciated!! Thanks!!
Way easier to understand than my textbook, thanks Professor Dave!
Your explanations are otherworldly!
So you don't understand.
The animation in revolving the function is so satisfying.
ive been lost for 2 semesters your video was the only one that made sense to me thank you for explaining it amazing
Professor Dave deserves every good thing that comes to him in life.
I'm only in 8th form of school, i don't understand some things ,but i really like your videos. Thank you!
do it chronologically
Work hard, you'll eventually understand. I was in 7th grade when i was starting to wonder, why.
Keep going on, you can do it.
Nameless same I’m in 7th right now though
Im a senior and this shits rly hard lmao
Your work professor is marvellous I love it thank you so much, please add more work for calculating volumes,
I don't get how the average radius magically becomes r. Its clearly a value greater than r1 and less than r2 but you show it to be equal to r2 when you claim that 2*pi*r is the circumference of the outer circle at 4:07.
I was wondering the same thing
I'd like to learn that kind of magic too
they just make the difference so small that (r1-r2) which is delta r becomes dx in the limit, and when the difference is so small (r1+r2)/2 magically becomes r cus r1 almost equal to r2. dunno if this is correct.
Your video had great explanation, and I really liked that you had 3 dimensional models.
The most to the point (time saving) and easy to understand teaching method. THANK YOU!!!!!!!!🙌🙌😍😍
U cleared my concept about volume using solid revolution.
Thanku Sir
Thank you sir for your dedication and for making this free! 🙏
Never thought i'd be coming to this channel for homework help and not watching you absolutely wreck the discovery institute but here we are i guess, thanks man
thank you professor Dave for making these videos
Surviving through your explanations this quarter 🙏 ... our professional did not care if the students understood
professor dave saves my grade once again
5:00 is the first time I heard we integrate from 0 to 2.
Anyway, rn I don't get how Δr disappeared at 4:25, or how it became dx.
I'll have to think that one through.
In the future feel free to ask how, me if I'll find out I can explain
thats what im wondering...
I think it is because:
m = f'(x) = dy/dx = (f(x+a) - f(x)) / (x + h - x) = (f(x+a) - f(x)) / a
so dx = a = x2 - x1 and in calculus as far as I know normally dx = Δx while dy != Δy.
In this video r = x and by that dx = r2 - r1 which would be Δr.
At least I think that is the reason why. If I'm wrong you may correct me.
How will the formula change if the function revolved about a line other than y or x axis? Say it revolved about x=4.
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best video ever. I love you so much. U r literally saving the world by nurturing future scholars
This 7 minute video taught me more than a 2.5 hour lecture from my professor.
per my understanding 2pixh is a surface area of cillinder at point x and by integration through x basically we get area integrated through x and hence the volume of shape. That's how I see it.
Greatest professor in human history
4:07 why does C have a circumference of a circle with the greater radius instead of the average radius? (In the graphic)
Nicely explained the main concept.
Thank you professor Dave for your great work.
suppose that a cylindrical can is designed to have a radius of 1 inc . and a height of 5 inc, but that the radius and height are off by amounts dr=+0.03 and dh=-0.01. estimate the resulting absolute change in the volume of a can
perfect, thank you so much
while findig the formula for the volume of the cilincrical shell, I don't get how could 2*pi*r be the circumference of the cilinder, since r2 is obviously the radius for its base circle, not the avarege radius. am I not noticing something?
the circumference of a circle is pi times its diameter, which is twice its radius.
@@ProfessorDaveExplains right but you said r was the average radius, not the bigger one
It might be though the difference is approaching to 0 when you deal with infinitessimals
@@szilike_10 still didn't get it. Did you understand?
Same doubt
why is it when calculating volume of a revolution with circles, we integrate a circle without a thichness onlly pi*r^2 but with cylindrical shells we add a thickness, why can't we use 2pi*r*h?
I just realised the thickness (delta r) may just be dx but I am not sure
In comprehension , why you put integrated version of the f(x) in V formula .
Please explain 1nd and 2nd step
ty for helping me more than my college professors
Tried Solving the comprehension check Problem-
Wasted more than 2 hours, but couldn't really solve after all-i understand the concept but i just got stuck at arithmetic shit at the end-
These kind of things really mess with your motivation.......
Should i give up?
Very good explanation
Thanks Jesus, I knew you would help me some day!
Professor Dave you are the best.
Why are we able to swap out the average of r for r?
How the equation was written at 4:25 ?
I understand that we just call the average value of the radii 'r', but why did we call (r2-r1) delta r? I did not get that part at all.
r2 - r1 is how much we have moved along the x-axis to get the next cylindrical shell, so this is delta x. When we integrate, we make each shell infinitely thin and so we set this delta x to approach 0 in the limit.
You can think of this approach as those old style pirate telescopes that have smaller cylinders inside larger cylinders and can extend or collapse. Only here you've got infinitely many of them and each one is infinitely thin. The height of the cylinders follows the function. When you collapse them all inside each other, you get the whole bundt cake. When you integrate you're summing up the infinitesimal volumes of the cylinders by going along the x-axis (moving r2-r1 by an infestimal amount each time). You only need to integrate from 0 to 2 because each change in x gives you the whole cylinder shell. no need to worry about -2 to 0, that volume is already included.
Is this not on a calculus textbook??? I can’t find it anywhere. Maybe I’m looking wrong
Sir plz also make a tutorial about physics or astronomy.. waiting for it.
From the last 1 year. Only math series is going on
buddy, i've done like 60 tutorials on physics already! astronomy is coming very soon, i'm filming it now.
This bundt cake looking thing cracked me up lol
Why u took mean radius 'r' in total volume
At the limit we set Delta r to equal 0, and so r2-r1=0. At that point r2=r1=r and r2 + r1 just equals 2r. No need to bring up the average even, just keep in mind that at the limit we're dealing with infinitesimals.
Can I integrate the surface area of a cylinder without the top and bottom area to get the solution
Im so amazed I actually got it correct thank you
THE ABSOLUTE BEST
Can anyone explain how he got the range 0 to 2
Because it is when the function crosses x-axis again
You can see in the beginning he took only a certain region to "wrap" around the y axis and that region varied from x=0 to x=2.
{ equate f(x) to zero and we get two roots ..0 and 2 }
@@MakeupByAnanyaSharma Thank you!
How did u get the 0 to 2 interval?
Because we are trying to find the volume generated by rotating the curve between x=0 and x=2.
isnt it wrong in 4min mark, the line C should be in between r2 and r1
4:27 where does delta r go?
Thank you professor Dave
Is the solution still the same even if there is another given like y=1. Does it matter in solving this.
I guess it's only the border of the integration
thanks mr
Why did you evaluate from 0 to 2
Brilliant..!
How did you get the radius please
Thank you!
Thank you
Thank you sir
Dave is the goat
thanks for the video
it is very useful thx
you deserve more subscribers!
Ikr I was thinking the same
Where is the thickness in the volume formula when you integrated it?
It’s just dx.
Where did you get the limits of integration in the main example shown?
Same thought, I think he forgot to mention that it was a made up example.
How tf did r1+r2/2=r???
U are the best and i hope u know that .
best explanation on this topic, goooood shit
Can please u expalin how can we take value of a and b if it is not mention in question
You find the zeros of the function. Set the function equal to 0 and solve for all values of x, then look for the two positive values and see if they fall within the bound you're given.
Hope this helps!
thank you professsorrrrrrrrrrrrrrrrr
why make a value from 0 to 2 how did it come
solve for y=0 and find the x intercepts
thank you, Mathematixcs Jesus
Why didn't dy in place of dx?
Because we used the technique of cylindrical shells we don't have to make things in terms of y
I thought that you were dealing with a cones shaped curve
uhhhh are the bounds the x intercepts or y intercepts
thanks math jesus
soo.. the volume of big jello
Hlo
00:09 here you go
h
Istg old professors needs to retire if they can't help their students 😒
why you add +1 on the "2x³- x⁴" that it has been already a power rule?
Thank you so much