Integrals and Volumes by Cylindrical Shells: Example 1

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(Edited)

In this video I go over an example on finding the volume by using the method of cylindrical shells. The example involves solving the volume of the solid formed by rotating the region bounded by the functions y = 2x2 – x3 and y = 0.


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Integrals and Volumes by Cylindrical Shells: Example 1

Integrals Cylinderical Shells Example 1.jpeg

Example:

Find the volume of the solid obtained by rotating about the y-axis the region bounded by y = 2x2 – x3 and y = 0.

Solution:

Using the Cylindrical Shells method is easier than the washer method in this example because we do not have to solve for the local maximum value or write x in terms of y.

But in later examples I will show that using the washer method and other methods from my earlier videos is actually easier than the cylindrical shells method.



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I did not have such advanced things on one variable analysis courses. I had calculation of some volumes or lengths similar to this on multivariable calculus - Pappus-Guldin, lemniscates, cardioids, etc, such things. I think background of physics is really important at analysis - now I see that my fellows who went further into analysis, differential equations are mostly those who study also a little bit of physics.
What is BitChute? Is it worth to publish both on DTube and BitChute?
Finally, I have a question: Please look at problems 29-33 from that list and tell me if You can do any of them:
https://www.mimuw.edu.pl/~leszekp/dydaktyka/AC/AC-problems-2019.pdf
Thanks, have a nice day.

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Thanks for the info. I have a simple head-down approach towards learning, so I just go over whatever my calculus book covers, even if it takes many hours digging through the internet and my brain to solve what they left out haha.

Those problems are the first time I've been aware of "weighted integrals" so I can't be of help as of right now. In the future I may cover them.

Right now, I am nearing finishing my calculus book, and I may do linear algebra after words. Will look to go over concepts faster so I can cover more topics.

As for BitChute, I believe it is a good bet to publish there along with DTube. Right now it looks like BC is growing at a faster pace: https://www.similarweb.com/website/bitchute.com?competitors=d.tube

In the future, I believe there will be a protocol standard(s) that allow for publishing content just once with a unique ID that can be viewed on all or specific platforms you choose. But for now, I just publish on YT, BC, DT to expand my reach and lower the risk of any one platform.

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