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### Section 4-5 : 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