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### Section 6-4 : Volume With Cylinders

For problems 1 – 14 use the method cylinders 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 $$\displaystyle y = \frac{1}{x}$$, $$\displaystyle y = \frac{1}{3}$$ and $$\displaystyle x = \frac{1}{2}$$ about the $$y$$-axis.
10. Rotate the region bounded by $$\displaystyle y = \frac{1}{x}$$, $$\displaystyle y = \frac{1}{3}$$ and $$\displaystyle x = \frac{1}{2}$$ about the $$x$$-axis.
11. Rotate the region bounded by $$y = 6 - 2x$$, $$y = 3 + x$$ and $$x = 3$$ about the $$y$$-axis.
12. Rotate the region bounded by $$y = 6 - 2x$$, $$y = 3 + x$$ and $$y = 6$$ about the $$x$$-axis.
13. Rotate the region bounded by $$y = {x^2} - 6x + 11$$ and $$y = 6$$ about the $$y$$-axis.
14. Rotate the region bounded by $$x = {y^2} - 8y + 19$$ and $$x = 2y + 3$$ about the $$x$$-axis.
15. Use the method of cylinders 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 $$y = -4$$
16. Use the method of cylinders to determine the volume of the solid obtained by rotating the region bounded by $$x = {y^3}$$, $$x = 8$$ and the $$x$$-axis about the
1. line $$x = 10$$
2. line $$x = -3$$
1. line $$y = 3$$
2. line $$y = -4$$
17. Use the method of cylinders 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 $$y = 7$$
2. line $$y = -2$$
18. Use the method of cylinders 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 $$y = -3$$
19. Use the method of cylinders to determine the volume of the solid obtained by rotating the region bounded by $$y = 4 + 3{{\bf{e}}^{ - x}}$$, $$y = 2$$, $$x = \frac{1}{2}$$ and $$x = 3$$ about the
1. line $$x = 5$$
2. line $$\displaystyle x = \frac{1}{4}$$
3. line $$x = - 1$$
20. Use the method of cylinders to determine the volume of the solid obtained by rotating the region bounded by $$x = {y^2} - 8y + 19$$ and $$x = 2y + 3$$ about the
1. line $$y = 9$$
2. line $$y = 1$$
3. line $$y = -3$$
21. Use the method of cylinders to determine the volume of the solid obtained by rotating the region bounded by $$y = 5 + \sqrt {x - 3}$$, $$y = 5$$ and $$x = 4$$ about the
1. line $$x = 9$$
2. line $$x = 2$$
3. line $$x = -1$$
22. Use the method of cylinders to determine the volume of the solid obtained by rotating the region bounded by $$y = - {x^2} - 10x + 6$$ and $$y = 2x + 26$$ about the
1. line $$x = 2$$
2. line $$x = -1$$
3. line $$x = =14$$
23. Use the method of cylinders to determine the volume of the solid obtained by rotating the region bounded by $$x = {y^2} - 10y + 27$$ and $$x = 11$$ about the
1. line $$y = 10$$
2. line $$y = 1$$
3. line $$y = -3$$
24. Use the method of cylinders to determine the volume of the solid obtained by rotating the region bounded by $$y = 2{x^2} + 1$$, $$y = 7 - x$$, $$x = 3$$ and $$\displaystyle x = \frac{3}{2}$$ about the
1. line $$x = 6$$
2. line $$x = 1$$
3. line $$x = -2$$