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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.

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Volumes Using Cross-Sections

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  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

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