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### Section 6-3 : Volume With Rings

For each of the following problems use the method of 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 $$y = \sqrt x$$, $$y = 3$$ and the $$y$$-axis about the $$y$$-axis. Solution
2. Rotate the region bounded by $$y = 7 - {x^2}$$, $$x = - 2$$, $$x = 2$$ and the $$x$$-axis about the $$x$$-axis. Solution
3. Rotate the region bounded by $$x = {y^2} - 6y + 10$$ and $$x = 5$$ about the $$y$$-axis. Solution
4. Rotate the region bounded by $$y = 2{x^2}$$ and $$y = {x^3}$$ about the $$x$$-axis. Solution
5. Rotate the region bounded by $$y = 6{{\bf{e}}^{ - 2x}}$$ and $$y = 6 + 4x - 2{x^2}$$ between $$x = 0$$ and $$x = 1$$ about the line $$y = - 2$$. Solution
6. Rotate the region bounded by $$y = 10 - 6x + {x^2}$$, $$y = - 10 + 6x - {x^2}$$, $$x = 1$$and $$x = 5$$ about the line $$y = 8$$. Solution
7. Rotate the region bounded by $$x = {y^2} - 4$$ and $$x = 6 - 3y$$ about the line $$x = 24$$. Solution
8. Rotate the region bounded by $$y = 2x + 1$$, $$x = 4$$ and $$y = 3$$ about the line $$x = - 4$$. Solution