The Shell Method | Calculus 2 Lesson 4 - JK Math
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- čas přidán 5. 08. 2024
- How to Use The Shell Method To Calculate Volume (Calculus 2 Lesson 4)
In this video we look at how to use definite integrals to calculate the volume of solids of revolution using the shell method. We discuss how to use the method when a region is revolved around the x-axis to form a solid, when a region is revolved around the y-axis to form a solid, and when the region is revolved around a line other than the x- or y-axis to form a solid. Additionally, we compare the shell method to the disk and washer methods and when each method is preferable.
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This series is designed to help students understand the concepts of Calculus 2 at a grounded level. No long, boring, and unnecessary explanations, just what you need to know at a reasonable and digestible pace, with the goal of each video being shorter than the average school lecture!
Calculus 2 requires a solid understanding of calculus 1, precalculus, and algebra concepts and techniques. This includes limits, differentiation, basic integration, factoring, equation manipulation, trigonometric functions, logarithms, graphing, and much more. If you are not familiar with these prerequisite topics, be sure to learn them first!
Video Chapters:
0:00 The Shell Method (y-axis)
6:15 The Shell Method (x-axis)
9:14 Summary of Formulas
10:46 How to Adjust Height When Between Two Curves
12:34 Example 1 - y=x^3, x=1, y=0 around y-axis
17:35 Example 2 - y=x^2, x=1, y=0 around x-axis
25:02 How to Adjust Radius When Revolving Around Other Lines
28:50 Example 3 Part 1 - y=x, y=sqrt(x) around y=1
32:41 Example 3 Part 2 - y=x, y=sqrt(x) around y=-1
34:55 Example 3 Part 3 - y=x, y=sqrt(x) around x=1
37:23 Example 3 Part 4 - y=x, y=sqrt(x) around x=-1
39:19 Comparison to Disk/Washer Method
41:02 Example 4 - When Shell Method is Preferable
46:30 Outro
📝 Examples Video: • The Shell Method Examp...
⏩ Next Lesson: • Volume Using Known Cro...
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📺 Calculus 1 Playlist: • Calculus 1
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i think there is a mistake at the time 46:15 the therms should in y because the revolution about y axis by disc method
Good catch! My apologies on that mistake. You are correct, since we are revolving around the y-axis, the integral should be entirely in terms of y for the disk/washer method. Can't believe I didn't catch that when I edited this video, especially since I correctly set up the first integral in terms of y, but not the second. To fix the mistake, solve for x in the function y=x^2+1 to get x=√(y-1) as the inner radius and substitute that in for where I put x^2+1. And then of course, there should be a dy instead of dx at the end of the integral. Thanks for pointing this out, again, my apologies on this mistake and any confusion it may have caused.
@JKMath you are great go on sir
Thanks! Watched khan academy first, and it finally clicked with this video
This was amazing, thank you so much!!
You're welcome! Glad the video could help :)
Why so less views 💀 on a masterpiece
Just thought I'd point out that you don't even need to use an integral to find the area of the bottom part - it's a rectangle!
This is a question on average radius r at 4:30. Normally r is just equal to x. Instead of x, can we just use (a+b)/2 which is the actual average radius? Conceptually, we can take the area bounded by f(x) from a to b multiplied by the thickness 2pi(r) where r = (a+b)/2.
No, the radius always needs to be defined in terms of x, it will not be constant. I see why you would think it could be (a+b)/2, but remember that in that diagram I drew, that was just ONE cylinder that is representative of many cylinders we are using to find the volume using the shell method. So each cylinder will have a different average radius. That's why we need to use x, as the average radius of each cylinder will be dependent on where those cylinders are located along the x-axis. You can test this out with the example problems in this video, if you change the radius from x to (a+b)/2 you will get a completely different answer, and it will be an incorrect answer. Hope this helps!
@@JKMath
Thanks for the reply.
try doing black background with white writing! Better on eyes and easier to focus
Thanks for the feedback! I currently do not have any plans to change the style of my videos, however I could look into releasing alternate versions in the future with inverted colors, which would make the background black and the writing white. I’ll keep it in mind. Hope you still find the videos to be helpful regardless! :)
the "enhancer for youtube" extension available on chromium and firefox has a setting that allows you to invert the colors yourself, it's nice for these kinds of videos.