1 / 44

Volumes Using Cross-Sections

Volumes Using Cross-Sections. Solids. Solids not generated by Revolution. Solids of Revolution. Examples: Classify the solids. Volumes Using Cross-Sections. Solids. Solids not generated by Revolution. Solids of Revolution. Volumes Using Cross-Sections. Solids of Revolution.

jblackston
Télécharger la présentation

Volumes Using Cross-Sections

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Volumes Using Cross-Sections Solids Solids not generated by Revolution Solids of Revolution Examples:Classify the solids

  2. Volumes Using Cross-Sections Solids Solids not generated by Revolution Solids of Revolution

  3. Volumes Using Cross-Sections Solids of Revolution Volumes Using Cross-Sections Volumes Using Cylindrical Shells Sec(6.1) Sec(6.2) The Washer Method The Disk Method

  4. VOLUMES 1 The Disk Method Strip with small width generate a disk after the rotation

  5. VOLUMES 1 The Disk Method Several disks with different radius r

  6. VOLUMES

  7. VOLUMES Intersection point between L, curve 1 Disk cross-section x step1 Graph and Identify the region Draw a line (L) perpendicular to the rotating line at the point x step2 Rotate this line. A circle is generated step3 Intersection point between L, rotating axis Find the radius r of the circe in terms of x step4 step7 The volume is given by Now the cross section Area is step5 Specify the values of x step6

  8. VOLUMES

  9. VOLUMES

  10. Volumes Using Cross-Sections Volumes Using Cross-Sections The Washer Method Sec(6.1) The Disk Method

  11. Volumes Using Cross-Sections Volumes Using Cross-Sections The Washer Method Sec(6.1) The Disk Method Examples:Classify

  12. Volumes Using Cross-Sections Volumes Using Cross-Sections The Washer Method Sec(6.1) The Disk Method Examples:Classify

  13. VOLUMES Volume = Area of the base X height

  14. VOLUMES

  15. VOLUMES 2 The washer Method If the cross-section is a washer ,we find the inner radius and outer radius

  16. VOLUMES 2 The washer Method Intersection point between L, boundary step1 Graph and Identify the region step2 Intersection point between L, boundary Draw a line perpendicular to the rotating line at the point x step3 Rotate this line. Two circles created step4 Find the radius r(out) r(in) of the washer in terms of x step5 Now the cross section Area is step6 Specify the values of x The volume is given by Intersec pt between L, rotation axis step7

  17. VOLUMES T-102

  18. VOLUMES Example: Find the volume of the solid obtained by rotating the region enclosed by the curves y=x and y=x^2 about the line y=2 . Find the volume of the resulting solid.

  19. VOLUMES 3 The Disk Method (about y-axis) If the cross-section is a disk, we find the radius of the disk (in terms of y ) and use

  20. VOLUMES The Disk Method (about y-axis) step1 Graph and Identify the region Draw a line (L) perpendicular to the rotating line at the point y step2 Rotate this line. A circle is generated step3 Find the radius r of the circe in terms of y step4 Now the cross section Area is step5 Specify the values of y step6 The volume is given by step7

  21. VOLUMES 4 The Washer Method (about y-axis or parallel) Example: The region enclosed by the curves y=x and y=x^2 is rotated about the line x= -1 . Find the volume of the resulting solid.

  22. VOLUMES 4 washer cross-section y step1 Graph and Identify the region step2 Draw a line perpendicular to the rotating line at the point y step3 Rotate this line. Two circles created step4 Find the radius r(out) r(in) of the washer in terms of y step5 Now the cross section Area is step6 Specify the values of y The volume is given by step7

  23. VOLUMES SUMMARY: The solids in all previous examples are all called solids of revolution because they are obtained by revolving a region about a line. Rotated by a line parallel to x-axis ( y=c) solids of revolution Rotated by a line parallel to y-axis ( x=c) NOTE: The cross section is perpendicular to the rotating line Cross-section is DISK solids of revolution Cross—section is WASHER

  24. VOLUMES BY CYLINDRICAL SHELLS Remarks rotating line Parallel to x-axis CYLINDRICAL SHELLS (6.2) rotating line Parallel to y-axis Remarks rotating line Parallel to x-axis Using Cross-Section(6.1) rotating line Parallel to y-axis Cross-section is DISK Cross—section is WASHER SHELL Method

  25. VOLUMES parallel to x-axis SHELLS Cross-Section parallel to y-axis

  26. VOLUMES BY CYLINDRICAL SHELLS T-131 Remark: before you start solving the problem, read the choices to figure out which method you use

  27. T-111

  28. VOLUMES T-102

  29. Volumes Using Cross-Sections Solids Solids not generated by Revolution Solids of Revolution

  30. Volumes Using Cross-Sections Example: 2 The base of a solid is bounded by the curve y = x /2 and the line y =2. If the cross-sections of the solid perpendicular to the y-axis are squares, then find the volume of the solid Cross-sections: Base: is bounded by the curve and the line y =2 If the cross-sections of the solid perpendicular to the y-axis are squares

  31. VOLUMES Jonathan Mitchell

  32. VOLUMES Example: The base of a solid is bounded by the curve and the line x =9. If the cross-sections of the solid perpendicular to the x-axis are semicircle, then find the volume of the solid Base: is bounded by the curve and the line x =9 Cross-sections: If the cross-sections of the solid perpendicular to the x-axis are semicircle

  33. VOLUMES Example: The base of a solid is bounded by the curve and the line x =9. If the cross-sections of the solid perpendicular to the x-axis are semicircle, then find the volume of the solid Base: is bounded by the curve and the line x =9 Cross-sections: If the cross-sections of the solid perpendicular to the x-axis are semicircle

  34. VOLUMES Example: The base of a solid is bounded by the curve and the line x =9. If the cross-sections of the solid perpendicular to the x-axis are semicircle, then find the volume of the solid Base: is bounded by the curve and the line x =9 Cross-sections: If the cross-sections of the solid perpendicular to the x-axis are semicircle

  35. VOLUMES Example: The base of a solid is bounded by the curve and the line y = 0 from x=0 to x=pi. If the cross-sections of the solid perpendicular to the x-axis are equilatral triangles. Base: is bounded by the curve and the line y =0 Cross-sections: If the cross-sections of the solid perpendicular to the x-axis are semicircle

  36. VOLUMES Example: The base of a solid is bounded by the curve and the line y = 0 from x=0 to x=pi. If the cross-sections of the solid perpendicular to the x-axis are equilatral triangles. Base: is bounded by the curve and the line y =0 Cross-sections: If the cross-sections of the solid perpendicular to the x-axis are equilatral triangles

  37. Volumes Using Cross-Sections step1 Graph and Identify the region ( graph with an angle) Example: 2 Draw a line (L) perpendicular to the x-axis (or y-axis) at the point x (or y), (as given in the problem) The base of a solid is bounded by the curve y = x /2 and the line y =2. If the cross-sections of the solid perpendicular to the y-axis are squares, then find the volume of the solid step2 Find the length (S)of the segment from the two intersection points with the boundary step3 Cross-sections: If the cross-sections of the solid perpendicular to the y-axis are squares Cross-section type: Square S = side length Semicircle  S = diameter Equilatral  S = side length step4 Cross-section type: Square Semicircle  Equilatral  step4 Specify the values of x step6 The volume is given by step7

  38. VOLUMES step1 Graph and Identify the region ( graph with an angle) Example: Draw a line (L) perpendicular to the x-axis (or y-axis) at the point x (or y), (as given in the problem) The base of a solid is bounded by the curve and the line x =9. If the cross-sections of the solid perpendicular to the x-axis are semicircle, then find the volume of the solid step2 Cross-sections: Find the length (S)of the segment from the two intersection points with the boundary step3 If the cross-sections of the solid perpendicular to the x-axis are semicircle Cross-section type: Square  S = side length Semicircle  S = diameter Equilatral  S = side length step4 Cross-section type: Square  Semicircle  Equilatral  step4 Specify the values of x step6 The volume is given by step7

  39. VOLUMES T-102

  40. VOLUMES T-122

  41. VOLUMES T-092

  42. VOLUMES

  43. T-132

  44. T-132

More Related