Paul's Online Notes
Home / Calculus III / Multiple Integrals / Triple Integrals in Cylindrical Coordinates
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### 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