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### Section 15.7 : Triple Integrals in Spherical Coordinates

1. Evaluate $$\displaystyle \iiint\limits_{E}{{10xz + 3\,dV}}$$ where $$E$$ is the region portion of $${x^2} + {y^2} + {z^2} = 16$$ with $$z \ge 0$$. Solution
2. Evaluate $$\displaystyle \iiint\limits_{E}{{{x^2} + {y^2}\,dV}}$$ where $$E$$ is the region portion of $${x^2} + {y^2} + {z^2} = 4$$ with $$y \ge 0$$. Solution
3. Evaluate $$\displaystyle \iiint\limits_{E}{{3z\,dV}}$$ where $$E$$ is the region inside both $${x^2} + {y^2} + {z^2} = 1$$ and $$z = \sqrt {{x^2} + {y^2}}$$. Solution
4. Evaluate $$\displaystyle \iiint\limits_{E}{{{x^2}\,dV}}$$ where $$E$$ is the region inside both $${x^2} + {y^2} + {z^2} = 36$$ and $$z = - \sqrt {3{x^2} + 3{y^2}}$$. Solution
5. Evaluate the following integral by first converting to an integral in spherical coordinates. $\int_{{ - 1}}^{0}{{\int_{{ - \sqrt {1 - {x^{\,2}}} }}^{{\sqrt {1 - {x^{\,2}}} }}{{\int_{{\sqrt {6{x^{\,2}} + 6{y^{\,2}}} }}^{{\sqrt {7 - {x^{\,2}} - {y^{\,2}}} }}{{\,\,\,18y\,\,\,dz}}\,dy}}\,dx}}$ Solution