Question

Question: EvaluatedV, Where B is the ball with center the origin and radius 5.

Short answer

The value of the given triple integral is \(\frac{{{\rm{312500\pi }}}}{{\rm{7}}}\).

Step-by-step solution

Explanation of solution

The formula for triple integration in spherical coordinates:

E is a spherical wedge indicated by

\(E = \{ (\rho ,\theta ,\phi )\mid a \le \rho \le b,\alpha \le \theta \le \beta ,c \le \phi \le d\} \).

The region of integration can be defined in the following way using the given integral:

\({\rm{B = \{ (\rho ,}}\theta ,\phi )\mid 0 \le \rho \le {\rm{5,0}} \le \theta \le {\rm{2\pi ,0}} \le \phi \le {\rm{\pi \} }}\)

Using \({{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ + }}{{\rm{z}}^{\rm{2}}}{\rm{ = }}{{\rm{\rho }}^{\rm{2}}}\)

\(\int_0^\pi {\int_0^{2\pi } {\int_0^5 {{{\left( {{\rho ^2}} \right)}^2}} } } {\rho ^2}\sin \phi d\rho d\theta d\phi = \int_0^\pi {\int_0^{2\pi } {\int_0^5 {{\rho ^6}} } } \sin \phi d\rho d\theta d\phi \).

Calculation                      

\(\begin{array}{c}\int_0^\pi {\int_0^{2\pi } {\int_0^5 {{\rho ^6}} } } \sin \phi d\rho d\theta d\phi = \left. {\int_0^\pi {\int_0^{2\pi } {\left( {\frac{{{\rho ^7}}}{7}\sin \phi } \right)} } } \right|_0^5d\theta d\phi \\ = \int_0^\pi {\int_0^{2\pi } {\frac{{{5^7}}}{7}} } \sin \phi d\theta d\phi \end{array}\)

\(\begin{array}{c}\int_0^\pi {\int_0^{2\pi } {\int_0^5 {{\rho ^6}} } } \sin \phi d\rho d\theta d\phi {\rm{ = }}\left. {\frac{{{\rm{78125}}}}{{\rm{7}}}{\rm{\theta }}} \right|_{\rm{0}}^{{\rm{2\pi }}}\int_{\rm{0}}^{\rm{\pi }} {{\rm{sin}}} \phi {\rm{d}}\phi \\{\rm{ = }}\frac{{{\rm{156250\pi }}}}{{\rm{7}}}\int_{\rm{0}}^{\rm{\pi }} {{\rm{sin}}} \phi {\rm{d}}\phi \\{\rm{ = }}\left. {\frac{{{\rm{156250\pi }}}}{{\rm{7}}}{\rm{( - cos}}\phi {\rm{)}}} \right|_{\rm{0}}^{\rm{\pi }}\\{\rm{ = }}\frac{{{\rm{156250\pi }}}}{{\rm{7}}}{\rm{( - cos\pi + cos0)}}\end{array}\)

\(\begin{array}{c}\int_0^\pi {\int_0^{2\pi } {\int_0^5 {{\rho ^6}} } } \sin \phi d\rho d\theta d\phi {\rm{ = }}\frac{{{\rm{156250\pi }}}}{{\rm{7}}}{\rm{ \times 2}}\\{\rm{ = }}\frac{{{\rm{312500\pi }}}}{{\rm{7}}}\end{array}\)

Hence, the value of the given triple integral is \(\frac{{{\rm{312500\pi }}}}{{\rm{7}}}\).