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

1. Evaluate $$\displaystyle \iiint\limits_{E}{{4xy\,dV}}$$ where $$E$$ is the region bounded by $$z = 2{x^2} + 2{y^2} - 7$$ and $$z = 1$$. Solution
2. Evaluate $$\displaystyle \iiint\limits_{E}{{{{\bf{e}}^{ - {x^{\,2}} - {z^{\,2}}}}\,dV}}$$ where $$E$$ is the region between the two cylinders $${x^2} + {z^2} = 4$$ and $${x^2} + {z^2} = 9$$ with $$1 \le y \le 5$$ and $$z \le 0$$. Solution
3. Evaluate $$\displaystyle \iiint\limits_{E}{{z\,dV}}$$ where $$E$$ is the region between the two planes $$x + y + z = 2$$ and $$x = 0$$ and inside the cylinder $${y^2} + {z^2} = 1$$. Solution
4. Use a triple integral to determine the volume of the region below $$z = 6 - x$$, above $$z = - \sqrt {4{x^2} + 4{y^2}}$$ inside the cylinder $${x^2} + {y^2} = 3$$ with $$x \le 0$$. Solution
5. Evaluate the following integral by first converting to an integral in cylindrical coordinates. $\int_{0}^{{\sqrt 5 }}{{\int_{{ - \sqrt {5 - {x^{\,2}}} }}^{0}{{\int_{{{x^{\,2}} + {y^{\,2}} - 11}}^{{9 - 3{x^{\,2}} - 3{y^{\,2}}}}{{\,\,\,2x - 3y\,\,\,dz}}\,dy}}\,dx}}$ Solution