Calculus-Sec-10-4-Solution


  

  10.4  Areas and Lengths in Polar Coordinates   by SGLee - HSKim - JHLee

                                                                                                               http://youtu.be/CZ9SHMtqVy4

 

1-4. Find the area of the region that is bounded by the given curve and lies in the specified sector.

 

 1. ,  

            http://matrix.skku.ac.kr/cal-lab/cal-9-4-1.html 

 

 







 2. ,  

            http://matrix.skku.ac.kr/cal-lab/cal-9-4-2.html 

 

 

 .







 3. 

            http://matrix.skku.ac.kr/cal-lab/cal-9-4-3.html 

 

 







5-9. Find the area bounded by one loop of the given curve.

 

 5. 

            http://matrix.skku.ac.kr/cal-lab/cal-9-4-5.html 

 







6. 

           http://matrix.skku.ac.kr/cal-lab/cal-9-4-6.html 

 

 

.







 7. 

            http://matrix.skku.ac.kr/cal-lab/cal-9-4-7.html 

 

 







 8. 

            http://matrix.skku.ac.kr/cal-lab/cal-9-4-8.html 

 

 







 9. Find the area between a large loop and the enclosed small loop of the curve .

            http://matrix.skku.ac.kr/cal-lab/cal-9-4-9.html 

 

 

.







From figure, the area between a large loop and the enclosed small loop

of the curve  is .

 10. 

             http://matrix.skku.ac.kr/cal-lab/cal-9-4-10.html    

 

       







 11. 

             http://matrix.skku.ac.kr/cal-lab/cal-9-4-11.html 

 

 

.







12. 




13-15. Find the area of the region that lies inside the first curve and outside the second curve.

 

 13. 

              http://matrix.skku.ac.kr/cal-lab/cal-9-4-13.html  

 

 

      

      .







 14. 

              http://matrix.skku.ac.kr/cal-lab/cal-9-4-14.html 

 

 










 15. .

             http://matrix.skku.ac.kr/cal-lab/cal-9-4-15.html 

 

 

      Since  , then

      

  







16-17. Find all points of intersection of the given curves and find the area of the region that lies inside both curves.

 

 16. 

             http://matrix.skku.ac.kr/cal-lab/cal-9-4-16.html 

 

  

    

           

     







 17. 

            http://matrix.skku.ac.kr/cal-lab/cal-9-4-17.html 

 










18-21. Find the length of the polar curve.

 

18. 







 19. 




 20. Graph the curve  and find its length.

            http://matrix.skku.ac.kr/cal-lab/cal-9-4-20.html 

 

 

       

        .










21. Show that the area of the surface generated by rotating the polar curve   

     with  about the polar axis is .

 

      From http://mathworld.wolfram.com/SurfaceofRevolution.html, we have the formula

      .

      Since ,

      .

                                                                    

 

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