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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.

- Rotate the region bounded by \(y = 2{x^2}\), \(y = 8\) and the \(y\)-axis about the \(y\)-axis.
- Rotate the region bounded by \(y = 2{x^2}\), \(y = 8\) and the \(y\)-axis about the \(x\)-axis.
- Rotate the region bounded by \(y = 2{x^2}\), \(x = 2\) and the \(x\)-axis about the \(x\)-axis.
- Rotate the region bounded by \(y = 2{x^2}\), \(x = 2\) and the \(x\)-axis about the \(y\)-axis.
- Rotate the region bounded by \(x = {y^3}\), \(x = 8\) and the \(x\)-axis about the \(x\)-axis.
- Rotate the region bounded by \(x = {y^3}\), \(x = 8\) and the \(x\)-axis about the \(y\)-axis.
- Rotate the region bounded by \(x = {y^3}\), \(y = 2\) and the \(y\)-axis about the \(x\)-axis.
- Rotate the region bounded by \(x = {y^3}\), \(y = 2\) and the \(y\)-axis about the \(y\)-axis.
- 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.
- 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.
- Rotate the region bounded by \(y = 6 - 2x\), \(y = 3 + x\) and \(x = 3\) about the \(y\)-axis.
- Rotate the region bounded by \(y = 6 - 2x\), \(y = 3 + x\) and \(y = 6\) about the \(x\)-axis.
- Rotate the region bounded by \(y = {x^2} - 6x + 11\) and \(y = 6\) about the \(y\)-axis.
- Rotate the region bounded by \(x = {y^2} - 8y + 19\) and \(x = 2y + 3\) about the \(x\)-axis.
- 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
- line \(x = 3\)
- line \(x = -2\)

- line \(y = 11\)
- line \(y = -4\)

- 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
- line \(x = 10\)
- line \(x = -3\)

- line \(y = 3\)
- line \(y = -4\)

- 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
- line \(y = 7\)
- line \(y = -2\)

- 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
- line \(x = 12\)
- line \(x = 2\)
- line \(x = -1\)

- line \(y = 14\)
- line \(y = 1\)
- line \(y = -3\)

- 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
- line \(x = 5\)
- line \(\displaystyle x = \frac{1}{4}\)
- line \(x = - 1\)

- 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
- line \(y = 9\)
- line \(y = 1\)
- line \(y = -3\)

- 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
- line \(x = 9\)
- line \(x = 2\)
- line \(x = -1\)

- 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
- line \(x = 2\)
- line \(x = -1\)
- line \(x = =14\)

- 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
- line \(y = 10\)
- line \(y = 1\)
- line \(y = -3\)

- 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
- line \(x = 6\)
- line \(x = 1\)
- line \(x = -2\)