1. VOLUMES Volume = Area of the base X height

2. VOLUMES

3. For a solid S that isn’t a cylinder we first “cut” S into pieces and approximate each piece by a cylinder.

4. VOLUMES

5. If the cross-section is a disk, we find the radius of the disk (in terms of x ) and use 1 Disk cross-section x animation Rotating axis

6. VOLUMES 1 Disk cross-section x use your imagination

7. VOLUMES 1 Disk cross-section x use your imagination

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

9. VOLUMES

11. Volume = Area of the base X height

12. VOLUMES

13. If the cross-section is a washer,we find the inner radius and outer radius VOLUMES 2 washer cross-section x

14. VOLUMES step1 Graph and Identify the region step2 Draw a line perpendicular to the rotating line at the point x step3 Find the radius r(out) r(in) of the washer in terms of x step4 Now the cross section Area is step5 Specify the values of x step6 The volume is given by 2 washer cross-section x Intersection point between L, boundary Intersection point between L, boundry

15. VOLUMES T-102

16. 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.

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

18. VOLUMES 3 Disk cross-section y step1 Graph and Identify the region step2 Draw a line perpendicular to the rotating line at the point y step3 Find the radius r of the disk in terms of y step4 Now the cross section Area is step5 Specify the values of y step6 The volume is given by step2 Rewrite all curves as x = in terms of y

19. If the cross-section is a washer,we find the inner radius and outer radius VOLUMES 4 washer cross-section y 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.

20. 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 Find the radius r(out) and r(in) of the washer in terms of y step4 Now the cross section Area is step5 Specify the values of y step6 The volume is given by step2 Rewrite all curves as x = in terms of y

21. If the cross-section is a washer,we find the inner radius and outer radius VOLUMES 4 washer cross-section y T-102

22. 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

23. VOLUMES

24. Sec 6.2: VOLUMES not solids of revolution The solids in all previous examples are all called solids of revolution because they are obtained by revolving a region about a line. We now consider the volumes of solids that are not solids of revolution.

25. T-102 VOLUMES

26. T-092 VOLUMES