6.2   Volumes                                          by SGLee - HSKim, YJLim

                                                                                                              http://youtu.be/Fd4Mguf2dbU  

1-17. Calculate the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

1.  about the -axis.

    .

2. about the -axis.

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

    .

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

 3.  about the -axis.

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

     .

4.  about the -axis.

   http://matrix.skku.ac.kr/cal-lab/cal-6-2-4.html 

    .

5.  about the -axis.

    http://matrix.skku.ac.kr/cal-lab/cal-6-2-5.html 

     .

 6.  about the -axis.

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

    .

7.  about the -axis.

    .

8.  about the -axis.

    http://matrix.skku.ac.kr/cal-lab/cal-6-2-8.html 

    .

 9.  about the -axis.

           http://matrix.skku.ac.kr/cal-lab/cal-6-2-9.html 

    .

10.  about .

      http://matrix.skku.ac.kr/cal-lab/cal-6-2-10.html 

    .

11.  about .

    .

 12.  about .

     .

13.  about .

      http://matrix.skku.ac.kr/cal-lab/cal-6-2-13.html

     .

14.  about .

     http://matrix.skku.ac.kr/cal-lab/cal-6-2-14.html

    .

 15.  about .

             http://matrix.skku.ac.kr/cal-lab/cal-6-2-15.html 

    .

16.  about .

     http://matrix.skku.ac.kr/cal-lab/cal-6-2-16.html 

    .

17.  about .

     http://matrix.skku.ac.kr/cal-lab/cal-6-2-17.html

    .

18-23. Set up, but do not evaluate, an integral for the volume of the solid obtained

           by rotating the region bounded by the given curves about the specified line.

18.  about .

     .

19.  about .

     .

 20.  about .

             http://matrix.skku.ac.kr/cal-lab/cal-6-2-20.html 

    .

21.  about .

    .

 22.  about .

             http://matrix.skku.ac.kr/cal-lab/cal-6-2-22.html 

    .

23.  about .

     http://matrix.skku.ac.kr/cal-lab/cal-6-2-23.html 

    . 

24-27. Describe the solid of revolution whose volume is represented by the integral.

24. 

     http://matrix.skku.ac.kr/cal-lab/cal-6-2-24.html 

    ; about -axis.

 25. 

      http://matrix.skku.ac.kr/cal-lab/cal-6-2-25.html

    ; about -axis.

26. .

      http://matrix.skku.ac.kr/cal-lab/cal-6-2-26.html 

    ; about -axis.

    .

 27. .

      http://matrix.skku.ac.kr/cal-lab/cal-6-2-27.html

    ; about -axis.

    .

28-32. Find the volume of the described solid .

28. A right circular cone with height h and base radius r.

      .

29. A frustum of a right circular cone with height h, lower base radius R, and top radius r.

      .

 30. A tetrahedron with three mutually perpendicular faces and

             three mutually perpendicular edges with lengths 6cm, 8cm and 10cm.

       ,  .

31. The base of S is an elliptical region with boundary curve .

     Cross-sections perpendicular to the -axis are isosceles right triangles with the hypotenuse in the base.      

      .

32. A bowl is shaped like a hemisphere with diameter 40 cm. A ball with diameter 20 cm is placed in the bowl

     and water is poured into the bowl to a depth of  centimeters. Find the volume of water in the bowl.

       .

 33.  about the axis.

              http://matrix.skku.ac.kr/cal-lab/cal-6-3-18.html

              http://matrix.skku.ac.kr/cal-lab/sage-grapher-imp.html 

34-37. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

 34. , ; about the -axis.

 35. , ; about -axis

36. , ; about .

     http://math1.skku.ac.kr/home/pub/1142

        .

 37. , ; about .

              http://matrix.skku.ac.kr/cal-lab/cal-6-2-37.html 

    .

 38. Find the volume of the solids obtained by rotating the region bounded by

             the curves  and  about the following lines:

          (a) the -axis.

          (b) the -axis.

          (c) .

(a) 

     http://matrix.skku.ac.kr/cal-lab/cal-6-2-38.html 

(b)

(c)

39. Let  be the region in the first quadrant bounded by the curves  and .

     Calculate the following quantities.

   (a) The area of .

   (b) The volume obtained by rotating  about the -axis.

   (c) The volume obtained by rotating  about the -axis.

(a) 

     Answer =  which is

    http://matrix.skku.ac.kr/cal-lab/cal-6-2-39.html 

(b)

(c)

        .

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