Calculus III - Triple Integrals in Spherical Coordinates (Practice Problems)

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Section 15.7 : Triple Integrals in Spherical Coordinates

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
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
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
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
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