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### 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 $$xz$$-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 $$z = \sqrt {4y}$$.