Calculus-Sec-6-1-Solution


 6.1   Areas between Curves                         by SGLee - HSKim, YJLim

 

1-21. Find the area of the region, bounded by the given curves.

 

1. 

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


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2. 

   http://matrix.skku.ac.kr/cal-lab/cal-6-1-2.html 

 

 

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3. 

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

 

 

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4. 

 

     .             <= 적분구간 (0,1), integral 앞에 2*










 

5. 

 

 

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6. 

 

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7. 

 

     .                  <= 적분구간을 (0,1) 수정










8. 

 

 

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9. 

 

 

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10.      x=12  삭제

 

 

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11. 

 

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12. 

     http://matrix.skku.ac.kr/cal-lab/cal-6-1-12.html

 

 

     has solutions ,

    so  is positive where 







13. 

 

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14. 

 

 

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15. 

 

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16. 

 

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17. 

 

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18. 

 

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19. 

 

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20. 

 

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21.  y=cosx, y= (x-pi/2)*(x+pi/2) (which replace

)

 

      integral(cos(x) - (x-pi/2)*(x+pi/2), x, 0, pi/2) Answer: 1/12*pi^3 + 1 ).










22-23. Find the integral and interpret it as the area of a region.

 

22. 

 

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23. 

 

 

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24-25. Approximate the area of the region bounded by the given curves using the Midpoint Rule with .

 

24.                 http://wiki.sagemath.org/interact/calculus#Numerical_integrals_with_various_rules

 

 

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25. 

     http://matrix.skku.ac.kr/cal-lab/cal-6-1-25.html 

 

 

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26-29. Determine the area of the region bounded by the curves.

 

26. 

 

 

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27. 

 

 

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28. 

 

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29. 

 

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30. Find the area of region defined by the inequalities .

 

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31. Find the area enclosed by the loop of the curve with equation .

    (Tschirnhausen's cubic.) http://en.wikipedia.org/wiki/Tschirnhausen_cubic

 

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32. Find the area of the region bounded by the curve , the tangent line to this curve at , and the -axis.

     http://matrix.skku.ac.kr/cal-lab/cal-6-1-32.html

 

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33. Find the number  such that the line  divides the region bounded by

      the curves  and  into two regions with equal area.

 

 

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34. (a) Find the number  such that the line  bisects the area under the curve

        .

     (b) Find the number  such that the line  bisects the area in part (a).

 

 

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35. Find the values of  such that the area of the region enclosed by the parabolas  and  is 1944.

 

 

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      For  and    is another solution.

      Therefore .




36-40. Find the area of the region bounded by the given curves.

 

36. 

 

 

      Since ,

        

              

                                    




37. 

 

 

    Since 

      

                                




38. 

 

      Since 

      

                                    




39. 

 

     

                                          




40. 

     http://matrix.skku.ac.kr/cal-lab/cal-6-1-40.html 

 

 

     

                                            




                                                        

                                                                                  Cherry Blossom

 

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