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Applications of Integration

Applications of Integration. Volumes of Revolution Many thanks to http:// mathdemos.gcsu.edu / shellmethod /gallery/ gallery.html. Method of discs. Take this ordinary line. Revolve this line around the x axis. 2. 5. We form a cylinder of volume.

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Applications of Integration

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  1. Applications of Integration Volumes of Revolution Many thanks to http://mathdemos.gcsu.edu/shellmethod/gallery/gallery.html

  2. Method of discs

  3. Take this ordinary line Revolve this line around the x axis 2 5 We form a cylinder of volume

  4. We could find the volume by finding the volume of small disc sections 2 5

  5. If we stack all these slices… We can sum all the volumes to get the total volume

  6. To find the volume of a cucumber… we could slice the cucumber into discs and find the volume of each disc.

  7. The volume of one section: Volume of one slice =

  8. We could model the cucumber with a mathematical curve and revolve this curve around the x axis… 25 -5 Each slice would have a thickness dx and height y.

  9. The volume of one section: r = y value h = dx Volume of one slice =

  10. Volume of cucumber… Area of 1 slice Thickness of slice

  11. Take this function… and revolve it around the x axis

  12. We can slice it up, find the volume of each disc and sum the discs to find the volume….. Volume of one slice= Radius = y Area = Thickness of slice = dx

  13. Take this shape…

  14. Revolve it…

  15. Christmas bell…

  16. Divide the region into strips

  17. Form a cylindrical slice

  18. Repeat the procedure for each strip

  19. To generate this solid

  20. A polynomial

  21. Regions that can be revolved using disc method

  22. Regions that cannot….

  23. Model this muffin.

  24. Washer Method

  25. A different cake

  26. Slicing….

  27. Making a washer

  28. Revolving around the x axis

  29. Region bounded between y = 1, x = 0, y = 1 x = 0

  30. Volume generated between two curves y= 1

  31. Area of cross section.. f(x) g(x)

  32. dx

  33. Your turn: Region bounded between x = 0, y = x, 

  34. Region bounded between y =1, x = 1

  35. Region bounded betweeny = 1, x = 1

  36. Region bounded between

  37. Around the x axis- set it up

  38. Revolving shapes around the y axis

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