Question

Find the average distance from a point in a ball of radius a to its center.

Short answer

Therefore, the average distance of a point from the center is\(\frac{{{\rm{3a}}}}{{\rm{4}}}\).

Step-by-step solution

Average distance from origin

Average distance of a point from the origin is \(\frac{\iiint_{\text{E}}{\text{ }\!\!\rho\!\!\text{ }}\text{dV}}{\iiint_{\text{E}}{\text{d}}\text{V}}\)

Substitution

\(\begin{aligned} \iiint_E \rho dV &= \int_0^\pi {\int_0^{2\pi } {\int_0^a \rho } } \left( {{\rho ^2}\sin \phi } \right)d\rho d\theta d\phi \\ &= \left( {\int_0^\pi {\sin } \phi d\phi } \right)\left( {\int_0^{2\pi } d \theta } \right)\left( {\int_0^a {{\rho ^3}}d\rho } \right) \\ &= ( - \cos \phi )_0^\pi (\theta )_0^{2\pi }\left( {\frac{{{\rho ^4}}}{4}} \right)_0^a \\ &= (2)(2\pi )\left( {\frac{{{a^4}}}{4}} \right) \\ \end{aligned} \)

\(\iiint_E \rho dV = \pi {a^4}\)

Step 3:Using Volume of Sphere

is the Volume of the Sphere, which is\(\frac{{\rm{4}}}{{\rm{3}}}{\rm{\pi }}{{\rm{a}}^{\rm{3}}}\).

Average distance of a point from the center is \(\frac{{{\rm{\pi }}{{\rm{a}}^{\rm{4}}}}}{{\frac{{\rm{4}}}{{\rm{3}}}{\rm{\pi }}{{\rm{a}}^{\rm{3}}}}}{\rm{ = }}\frac{{{\rm{3a}}}}{{\rm{4}}}\).

Therefore, the average distance of a point from the center is \(\frac{{{\rm{3a}}}}{{\rm{4}}}\).