The Shell Method | Calculus 2 Lesson 4 - JK Math

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. 🖥️ Join My Membership Site: https://www.jkmathematics.com/plus 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: https://youtu.be/PDVzPFFHuF4 ⏩ Next Lesson: https://youtu.be/hnA9P2grADA 📺 Calculus 2 Playlist: https://youtube.com/playlist?list=PLHdRLeAbIZE7KQ922piyRIDAj30XYcOIu 📺 Calculus 1 Playlist: https://youtube.com/playlist?list=PLHdRLeAbIZE6krGDQthaQRpLZFLD4ob54 🌐 Visit My Website: https://www.jkmathematics.com Corrections: 46:15 I made a mistake here with the second integral. Since we are revolving around the y-axis, both integrals should be entirely in terms of y for the disk/washer method. To fix this 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. ⚡️Math Products I Recommend⚡️ Graphing Calculator: https://amzn.to/3XuUcB9 Non-Graphing Calculator: https://amzn.to/46hHJ7E Financial Calculator: https://amzn.to/3CRe9s4 Graphing Paper: https://amzn.to/46mAfAr College Rule Paper: https://amzn.to/3NOJaTN My Favorite Pencil: https://amzn.to/3NOFMIB My Favorite Erasers: https://amzn.to/3PumUQB Calculus Workbook: https://amzn.to/3r8SnNP ⚡️Textbooks I Use⚡️ Calculus 1 & 2: https://amzn.to/3PzfI5F Calculus 3: https://amzn.to/46rSy76 Financial Mathematics: https://amzn.to/3ppIVW0 ⚡️My Recording Equipment⚡️ iPad Air: https://amzn.to/3phEug1 Apple Pencil (Gen 2): https://amzn.to/3Nl45MM Tablet Desk Stand: https://amzn.to/4420rPh External Hardrive: https://amzn.to/44pkEyh (Commissions earned on qualifying purchases) Find me on social media: Facebook: https://www.facebook.com/jkmathematics Twitter: https://twitter.com/jk_mathematics Instagram: @jk_mathematics Found this video to be helpful? Consider giving this video a like and subscribing to the channel! Thanks for watching! Any questions? Feedback? Leave a comment! -Josh from JK Math #calculus Disclaimer: Please note that some of the links associated with the videos on my channel may generate affiliate commissions on my behalf. As an amazon associate, I earn from qualifying purchases that you may make through such affiliate links.