Calculus III - Triple Integrals in Cylindrical Coordinates (Assignment Problems)

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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 $a$ .