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

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

3. Volumes Using Cross-Sections Solids of Revolution Volumes Using Cross-Sections Volumes Using Cylindrical Shells Sec(6.2) Sec(6.1) The Disk Method The Washer 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. 1 Disk cross-section x step1 Graph and Identify the region step2 Draw a line (L) perpendicular to the rotating line at the point x step4 Find the radius r of the circe in terms of x step5 Now the cross section Area is step6 Specify the values of x step7 The volume is given by Intersection point between L, rotating axis Intersection point between L, curve step3 Rotate this line. A circle is generated

8. VOLUMES

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

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

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

13. VOLUMES Volume = Area of the base X height

14. VOLUMES

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

16. VOLUMES step1 Graph and Identify the region step2 Draw a line perpendicular to the rotating line at the point x 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 step7 The volume is given by 2 The washer Method Intersec pt between L, rotation axis Intersection point between L, boundary step3 Rotate this line. Two circles created

17. VOLUMES T-102

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

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

21. VOLUMES 4 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. The Washer Method (about y-axis or parallel)

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 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 step7 The volume is given by step3 Rotate this line. Two circles created

23. solids of revolution 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) Rotated by a line parallel to y-axis ( x=c) NOTE: The cross section is perpendicular to the rotating line solids of revolution Cross-section is DISK Cross—section is WASHER

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

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

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 of Revolution Solids Solids not generated by Revolution

30. Volumes Using Cross-Sections 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 Example: 2 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 Cross-sections:

31. VOLUMES Jonathan Mitchell

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

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

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

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

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

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

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

39. T-102 VOLUMES

40. T-122 VOLUMES

41. T-092 VOLUMES

43. T-132