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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.
- Rotate the region bounded by \(y = \sqrt x \), \(y = 3\) and the \(y\)-axis about the \(y\)-axis. Solution
- Rotate the region bounded by \(y = 7 - {x^2}\), \(x = - 2\), \(x = 2\) and the \(x\)-axis about the \(x\)-axis. Solution
- Rotate the region bounded by \(x = {y^2} - 6y + 10\) and \(x = 5\) about the \(y\)-axis. Solution
- Rotate the region bounded by \(y = 2{x^2}\) and \(y = {x^3}\) about the \(x\)-axis. Solution
- 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
- 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
- Rotate the region bounded by \(x = {y^2} - 4\) and \(x = 6 - 3y\) about the line \(x = 24\). Solution
- Rotate the region bounded by \(y = 2x + 1\), \(x = 4\) and \(y = 3\) about the line \(x = - 4\). Solution