Section 15.5 : 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 $xz$-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 $z = \sqrt {4y}$.