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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 15.6 : Triple Integrals in Cylindrical Coordinates
- Evaluate \( \displaystyle \iiint\limits_{E}{{8z\,dV}}\) where \(E\) is the region bounded by \(z = 2{x^2} + 2{y^2} - 4\) and \(z = 5 - {x^2} - {y^2}\) in the 1st octant.
- Evaluate \( \displaystyle \iiint\limits_{E}{{6xy\,dV}}\) where \(E\) is the region above \(z = 2x - 10\), below \(z = 2\) and inside the cylinder \({x^2} + {y^2} = 4\).
- Evaluate \( \displaystyle \iiint\limits_{E}{{9y{z^3}\,dV}}\) where \(E\) is the region between \(x = - \sqrt {9{y^2} + 9{z^2}} \) and \(x = \sqrt {{y^2} + {z^2}} \) inside the cylinder \({y^2} + {z^2} = 1\).
- Evaluate \( \displaystyle \iiint\limits_{E}{{x + 2\,dV}}\) where \(E\) is the region bounded by \(x = 18 - 4{y^2} - 4{z^2}\) and \(x = 2\) with \(z \ge 0\).
- Evaluate \( \displaystyle \iiint\limits_{E}{{x + 2\,dV}}\) where \(E\) is the region between the two planes \(2x + y + z = 6\) and \(6x + 3y + 3z = 12\) inside the cylinder \({x^2} + {z^2} = 16\).
- Evaluate \( \displaystyle \iiint\limits_{E}{{{x^2}\,dV}}\) where \(E\) is the region bounded by \(y = {x^2} + {z^2} - 4\) and \(y = 8 - 5{x^2} - 5{z^2}\) with \(x \le 0\).
- Use a triple integral to determine the volume of the region bounded by \(z = \sqrt {{x^2} + {y^2}} \), and \(z = {x^2} + {y^2}\) in the 1st octant.
- Use a triple integral to determine the volume of the region bounded by \(y = \sqrt {9{x^2} + 9{z^2}} \), and \(y = - 3{x^2} - 3{z^2}\) in the 1st octant.
- Use a triple integral to determine the volume of the region behind \(x = z + 3\), in front of \(x = - z - 6\) and inside the cylinder \({y^2} + {z^2} = 4\).
- Evaluate the following integral by first converting to an integral in cylindrical coordinates. \[\int_{{ - 4}}^{4}{{\int_{0}^{{\sqrt {16 - {y^{\,2}}} }}{{\int_{0}^{{6 + x}}{{\,\,\,6y{x^2}\,\,dz}}\,dx}}\,dy}}\]
- Evaluate the following integral by first converting to an integral in cylindrical coordinates. \[\int_{0}^{3}{{\int_{{ - \sqrt {9 - {x^{\,2}}} }}^{{\sqrt {9 - {x^{\,2}}} }}{{\int_{{ - \sqrt {2{x^{\,2}} + 2{y^{\,2}}} }}^{{6 + {x^{\,2}} + {y^{\,2}}}}{{\,\,\,15z\,\,dz}}\,dy}}\,dx}}\]
- Use a triple integral in cylindrical coordinates to derive the volume of a cylinder of height \(h\) and radius \(a\).