Pauls Online Notes
Home / Calculus III / Multiple Integrals / Triple Integrals
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.
Assignment Problems Notice
Please do not email me to get solutions and/or answers to these problems. I will not give them out under any circumstances nor will I respond to any requests to do so. The intent of these problems is for instructors to use them for assignments and having solutions/answers easily available defeats that purpose.

### Section 4-5 : Triple Integrals

1. Evaluate $$\displaystyle \int_{1}^{2}{{\int_{0}^{2}{{\int_{{ - 1}}^{1}{{2 + {z^2} - xy\,dz}}\,dx}}\,dy}}$$
2. Evaluate $$\displaystyle \int_{2}^{0}{{\int_{{{x^{\,2}}}}^{1}{{\int_{0}^{{x\,z}}{{{y^2} - 6z\,dy}}\,dz}}\,dx}}$$
3. Evaluate $$\displaystyle \int_{{ - 1}}^{2}{{\int_{0}^{1}{{\int_{0}^{{2z}}{{3x - \sqrt {1 + {z^2}} \,dx}}\,dz}}\,dy}}$$
4. Evaluate $$\displaystyle \iiint\limits_{E}{{12y\,dV}}$$ where $$E$$ is the region below $$6x + 4y + 3z = 12$$ in the first octant.
5. Evaluate $$\displaystyle \iiint\limits_{E}{{5{x^2}\,dV}}$$ where $$E$$ is the region below $$x + 2y + 4z = 8$$ in the first octant.
6. Evaluate $$\displaystyle \iiint\limits_{E}{{10{z^2} - x\,dV}}$$ where $$E$$ is the region below $$z = 8 - y$$ and above the region in the $$xy$$-plane bounded by $$y = 2x$$, $$x = 3$$ and $$y = 0$$.
7. Evaluate $$\displaystyle \iiint\limits_{E}{{4{y^2}\,dV}}$$where $$E$$ is the region below $$z = - 3{x^2} - 3{y^2}$$ and above $$z = - 12$$.
8. Evaluate $$\displaystyle \iiint\limits_{E}{{2y - 9z\,dV}}$$ where $$E$$ is the region behind $$6x + 3y + 3z = 15$$ front of the triangle in the $$xy$$-plane with vertices, in $$\left( {x,z} \right)$$ form :$$\left( {0,0} \right)$$, $$\left( {0,4} \right)$$and $$\left( {2,4} \right)$$.
9. Evaluate $$\displaystyle \iiint\limits_{E}{{18x\,dV}}$$ where $$E$$ is the region behind the surface $$y = 4 - {x^2}$$ that is in front of the region in the $$xy$$-plane bounded by $$z = - 3x$$, $$z = 2x$$ and $$z = 2$$.
10. Evaluate $$\displaystyle \iiint\limits_{E}{{20{x^3}\,dV}}$$ where $$E$$ is the region bounded by $$x = 2 - {y^2} - {z^2}$$ and $$x = 5{y^2} + 5{z^2} - 6$$.
11. Evaluate $$\displaystyle \iiint\limits_{E}{{6{z^2}\,dV}}$$ where $$E$$ is the region behind $$x + 6y + 2z = 8$$ that is in front of the region in the $$xy$$-plane bounded by $$z = 2y$$ and $$z = \sqrt {4y}$$.
12. Evaluate $$\displaystyle \iiint\limits_{E}{{8y\,dV}}$$ where $$E$$ is the region between $$x + y + z = 6$$ and $$x + y + z = 10$$ above the triangle in the $$xy$$-plane with vertices, in $$\left( {x,y} \right)$$ form : $$\left( {0,0} \right)$$, $$\left( {1,2} \right)$$ and $$\left( {1,4} \right)$$.
13. Evaluate $$\displaystyle \iiint\limits_{E}{{8y\,dV}}$$ where $$E$$ is the region between $$x + y + z = 6$$ and $$x + y + z = 10$$ in front of the triangle in the $$xy$$-plane with vertices, in $$\left( {x,z} \right)$$ form : $$\left( {0,0} \right)$$, $$\left( {1,2} \right)$$ and $$\left( {1,4} \right)$$.
14. Evaluate $$\displaystyle \iiint\limits_{E}{{8y\,dV}}$$ where $$E$$ is the region between $$x + y + z = 6$$ and $$x + y + z = 10$$ in front of the triangle in the $$xy$$-plane with vertices, in $$\left( {y,z} \right)$$ form : $$\left( {0,0} \right)$$, $$\left( {1,2} \right)$$ and $$\left( {1,4} \right)$$.
15. Use a triple integral to determine the volume of the region below $$z = 8 - y$$ and above the region in the $$xy$$-plane bounded by $$y = 2x$$, $$x = 3$$ and $$y = 0$$.
16. Use a triple integral to determine the volume of the region in the 1st octant that is below $$4x + 8y + z = 16$$.
17. Use a triple integral to determine the volume of the region behind $$6x + 3y + 3z = 15$$ front of the triangle in the $$xz$$-plane with vertices, in $$\left( {x,z} \right)$$ form :$$\left( {0,0} \right)$$, $$\left( {0,4} \right)$$and $$\left( {2,4} \right)$$.
18. Use a triple integral to determine the volume of the region bounded by $$y = {x^2} + {z^2}$$ and $$y = \sqrt {{x^2} + {z^2}}$$.
19. Use a triple integral to determine the volume of the region behind $$x + 6y + 2z = 8$$ that is in front of the region in the $$xy$$-plane bounded by $$z = 2y$$ and $$z = \sqrt {4y}$$.