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### Section 4-6 : Triple Integrals in Cylindrical Coordinates

1. 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.
2. Evaluate $$\displaystyle \iiint\limits_{E}{{6xy\,dV}}$$ where $$E$$ is the region above $$z = 2x - 10$$, below $$z = 2$$ and inside the cylinder $${x^2} + {z^2} = 4$$.
3. 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 $${y^2} + {z^2} = 1$$.
4. 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 $$z \ge 0$$.
5. 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 $${x^2} + {z^2} = 16$$.
6. 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 $$x \le 0$$.
7. 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.
8. 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.
9. Use a triple integral to determine the volume of the region below $$x = z + 3$$, above $$x = - z - 6$$ and inside the cylinder $${y^2} + {z^2} = 4$$.
10. Evaluate the following integral by first converting to an integral in cylindrical coordinates. $\int_{{ - 4}}^{4}{{\int_{0}^{{\sqrt {16 - {y^{\,2}}} }}{{\int_{0}^{{6 + x}}{{\,\,\,6y{x^2}\,\,dz}}\,dx}}\,dy}}$
11. Evaluate the following integral by first converting to an integral in cylindrical coordinates. $\int_{0}^{3}{{\int_{{ - \sqrt {9 - {x^{\,2}}} }}^{{\sqrt {9 - {x^{\,2}}} }}{{\int_{{ - \sqrt {2{x^{\,2}} + 2{y^{\,2}}} }}^{{6 + {x^{\,2}} + {y^{\,2}}}}{{\,\,\,15z\,\,dz}}\,dy}}\,dx}}$
12. Use a triple integral in cylindrical coordinates to derive the volume of a cylinder of height $$h$$ and radius $$a$$.