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Section 6-3 : Volume With Rings

For problems 1 - 16 use the method disks/rings to determine the volume of the solid obtained by rotating the region bounded by the given curves about the given axis.

1. Rotate the region bounded by $$y = 2{x^2}$$, $$y = 8$$ and the $$y$$-axis about the $$y$$-axis.
2. Rotate the region bounded by $$y = 2{x^2}$$, $$y = 8$$ and the $$y$$-axis about the $$x$$-axis.
3. Rotate the region bounded by $$y = 2{x^2}$$, $$x = 2$$ and the $$x$$-axis about the $$x$$-axis.
4. Rotate the region bounded by $$y = 2{x^2}$$, $$x = 2$$ and the $$x$$-axis about the $$y$$-axis.
5. Rotate the region bounded by $$x = {y^3}$$, $$x = 8$$ and the $$x$$-axis about the $$x$$-axis.
6. Rotate the region bounded by $$x = {y^3}$$, $$x = 8$$ and the $$x$$-axis about the $$y$$-axis.
7. Rotate the region bounded by $$x = {y^3}$$, $$y = 2$$ and the $$y$$-axis about the $$x$$-axis.
8. Rotate the region bounded by $$x = {y^3}$$, $$y = 2$$ and the $$y$$-axis about the $$y$$-axis.
9. Rotate the region bounded by $$y = \frac{1}{{{x^2}}}$$, $$y = 9$$, $$x = - 2$$, $$\displaystyle x = - \frac{1}{3}$$ about the $$y$$-axis.
10. Rotate the region bounded by $$y = \frac{1}{{{x^2}}}$$, $$y = 9$$, $$x = - 2$$, $$\displaystyle x = - \frac{1}{3}$$ about the $$x$$-axis.
11. Rotate the region bounded by $$y = 4 + 3{{\bf{e}}^{ - x}}$$, $$y = 2$$, $$\displaystyle x = \frac{1}{2}$$ and $$x = 3$$ about the $$x$$-axis.
12. Rotate the region bounded by $$x = 5 - {y^2}$$ and $$x = 4$$ about the $$y$$-axis.
13. Rotate the region bounded by $$y = 6 - 2x$$, $$y = 3 + x$$ and $$x = 3$$ about the $$x$$-axis.
14. Rotate the region bounded by $$y = 6 - 2x$$, $$y = 3 + x$$ and $$y = 6$$ about the $$y$$-axis.
15. Rotate the region bounded by $$y = {x^2} - 2x + 4$$ and $$y = x + 14$$ about the $$x$$-axis.
16. Rotate the region bounded by $$x = {\left( {y - 3} \right)^2}$$ and $$x = 16$$ about the $$y$$-axis.
17. Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by $$y = 2{x^2}$$, $$y = 8$$ and the $$y$$-axis about the
1. line $$x = 3$$
2. line $$x = - 2$$
1. line $$y = 11$$
2. line $$y = - 4$$
18. Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by $$x = {y^2} - 6y + 9$$ and $$x = - {y^2} + 6y - 1$$ about the
1. line $$x = 10$$
2. line $$x = -3$$
19. Use the method of disks/rings to determine the volume of the solid obtained by rotating the triangle with vertices $$\left( {3,2} \right)$$, $$\left( {7,2} \right)$$ and $$\left( {7,14} \right)$$ about the
1. line $$x = 12$$
2. line $$x = 2$$
3. line $$x = -1$$
1. line $$y = 14$$
2. line $$y = 1$$
3. line $$y = -3$$
20. Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by $$y = 4 + 3{{\bf{e}}^{ - x}}$$, $$y = 2$$, $$\displaystyle x = \frac{1}{2}$$ and $$x = 3$$ about the
1. line $$y = 7$$
2. line $$y = 1$$
3. line $$y = -3$$
21. Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by $$x = 3 + {y^2}$$ and $$x = 2y + 11$$ about the
1. line $$x = 23$$
2. line $$x = 2$$
3. line $$x = -1$$
22. Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by $$y = 5 + \sqrt x$$, $$y = 5$$ and $$x = 4$$ about the
1. line $$y = 8$$
2. line $$y = 2$$
3. line $$y = -2$$
23. Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by $$y = 10 - 2x$$, $$y = x + 1$$ and $$y = 7$$ about the
1. line $$x = 8$$
2. line $$x = 1$$
3. line $$x = -4$$
24. Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by $$y = - {x^2} - 2x - 5$$ and $$y = 2x - 17$$ about the
1. line $$y = 3$$
2. line $$y = -1$$
3. line $$y = -34$$
25. Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by $$x = - 2{y^2} - 3$$ and $$x = - 5$$ about the
1. line $$x = 4$$
2. line $$x = -2$$
3. line $$x = -9$$