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### Section 15.7 : Triple Integrals in Spherical Coordinates

1. Evaluate $$\displaystyle \iiint\limits_{E}{{4{y^2}\,dV}}$$ where $$E$$ is the sphere $${x^2} + {y^2} + {z^2} = 9$$.
2. Evaluate $$\displaystyle \iiint\limits_{E}{{3x - 2y\,dV}}$$ where $$E$$ is the region between the spheres $${x^2} + {y^2} + {z^2} = 1$$ and $${x^2} + {y^2} + {z^2} = 4$$ with $$z \le 0$$.
3. Evaluate $$\displaystyle \iiint\limits_{E}{{2yz\,dV}}$$ where $$E$$ is the region inside both $${x^2} + {y^2} + {z^2} = 16$$ and $$z = \sqrt {3{x^2} + 3{y^2}}$$ that is in the 1st octant.
4. Evaluate $$\displaystyle \iiint\limits_{E}{{{z^2}\,dV}}$$ where $$E$$ is the region between the spheres $${x^2} + {y^2} + {z^2} = 4$$ and $${x^2} + {y^2} + {z^2} = 25$$ and inside $$\displaystyle z = - \sqrt {\frac{1}{3}{x^2} + \frac{1}{3}{y^2}}$$.
5. Evaluate $$\displaystyle \iiint\limits_{E}{{5{y^2}\,dV}}$$ where $$E$$ is the portion of $${x^2} + {y^2} + {z^2} = 4$$ with $$x \le 0$$.
6. Evaluate $$\displaystyle \iiint\limits_{E}{{2 + 16x\,dV}}$$ where $$E$$ is the region between the spheres $${x^2} + {y^2} + {z^2} = 1$$ and $${x^2} + {y^2} + {z^2} = 4$$ with $$y \ge 0$$ and $$z \le 0$$.
7. Evaluate the following integral by first converting to an integral in cylindrical coordinates. $\int_{0}^{2}{{\int_{{ - \sqrt {4 - {x^{\,2}}} }}^{0}{{\int_{{\sqrt {5{x^{\,2}} + 5{y^{\,2}}} }}^{{\sqrt {9 - {x^{\,2}} - {y^{\,2}}} }}{{\,\,\,7x\,\,\,dz}}\,dy}}\,dx}}$
8. Evaluate the following integral by first converting to an integral in cylindrical coordinates. $\int_{{ - \sqrt 5 }}^{{\sqrt 5 }}{{\int_{0}^{{\sqrt {5 - {y^{\,2}}} }}{{\int_{{ - \sqrt {10 - {x^{\,2}} - {y^{\,2}}} }}^{{ - \sqrt {{x^{\,2}} + {y^{\,2}}} }}{{\,\,\,3x{z^2}\,\,\,dz}}\,dx}}\,dy}}$
9. Use a triple integral in spherical coordinates to derive the volume of a sphere with radius $$a$$.