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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 \(z = \sqrt {4y} \).