6.3    Volumes by Cylindrical Shells               by SGLee - HSKim, YJLim

                                                                                                        http://youtu.be/gNaKkA0UNHg   

1-18. Find the volume generated by rotating the region bounded by the given curves about the -axis using the method of cylindrical shells.

 1. ; about the axis.

          http://matrix.skku.ac.kr/cal-lab/cal-6-3-1.html 

         [You may refer to other open resources in http://math1.skku.ac.kr/pub/ ]

 2. ; about the axis.

            http://matrix.skku.ac.kr/cal-lab/cal-6-3-2.html 

 3. ; about the axis.

           http://matrix.skku.ac.kr/cal-lab/cal-6-3-3.html 

 4. ; about the axis.

           http://matrix.skku.ac.kr/cal-lab/cal-6-3-4.html 

 5.  about .

           http://matrix.skku.ac.kr/cal-lab/cal-6-3-6.html 

 6.  about .

           http://matrix.skku.ac.kr/cal-lab/cal-6-3-7.html 

 7.  about the axis.

           http://matrix.skku.ac.kr/cal-lab/cal-6-3-8.html 

 8.  about .

           http://matrix.skku.ac.kr/cal-lab/cal-6-3-9.html 

 9.  about the axis.

            http://matrix.skku.ac.kr/cal-lab/cal-6-3-15.html

 10.  about .

              http://matrix.skku.ac.kr/cal-lab/cal-6-3-16.html

                             
                           Korean International Math Olympiad Teams (2011)               South Korean International Math Olympiad Team (2012)

                                                                                                                                   http://www.imo-official.org/year_info.aspx?year=2012  

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