Section 15.5 : Triple Integrals

1. Evaluate $\displaystyle \int_{2}^{3}{{\int_{{ - 1}}^{4}{{\int_{1}^{0}{{4{x^2}y - {z^3}\,dz}}\,dy}}\,dx}}$ Solution
2. Evaluate $\displaystyle \int_{0}^{1}{{\int_{0}^{{{z^{\,2}}}}{{\int_{0}^{3}{{y\cos \left( {{z^5}} \right)\,dx}}\,dy}}\,dz}}$ Solution
3. Evaluate $\displaystyle \iiint\limits_{E}{{6{z^2}\,dV}}$ where $E$ is the region below $4x + y + 2z = 10$ in the first octant. Solution
4. Evaluate $\displaystyle \iiint\limits_{E}{{3 - 4x\,dV}}$ where $E$ is the region below $z = 4 - xy$ and above the region in the $xy$-plane defined by $0 \le x \le 2$, $0 \le y \le 1$. Solution
5. Evaluate $\displaystyle \iiint\limits_{E}{{12y - 8x\,dV}}$ where $E$ is the region behind $y = 10 - 2z$ and in front of the region in the $xz$-plane bounded by $z = 2x$, $z = 5$ and $x = 0$. Solution
6. Evaluate $\displaystyle \iiint\limits_{E}{{yz\,dV}}$ where $E$ is the region bounded by $x = 2{y^2} + 2{z^2} - 5$ and the plane $x = 1$. Solution
7. Evaluate $\displaystyle \iiint\limits_{E}{{15z\,dV}}$ where $E$ is the region between $2x + y + z = 4$ and $4x + 4y + 2z = 20$ that is in front of the region in the $yz$-plane bounded by $z = 2{y^2}$ and $z = \sqrt {4y}$. Solution
8. Use a triple integral to determine the volume of the region below $z = 4 - xy$ and above the region in the $xy$-plane defined by $0 \le x \le 2$, $0 \le y \le 1$. Solution
9. Use a triple integral to determine the volume of the region that is below $z = 8 - {x^2} - {y^2}$ above $z = - \sqrt {4{x^2} + 4{y^2}}$ and inside ${x^2} + {y^2} = 4$. Solution