Section 15.6 : 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