Question

Find the mass of a ball \({\rm{B}}\)given by \({{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ + }}{{\rm{z}}^{\rm{2}}} \le {{\rm{a}}^{\rm{2}}}\)if the density at any point is proportional to its distance from the z-axis.

Short answer

The Mass of the ball B is\(\frac{{{\rm{4\pi }}{{\rm{a}}^{\rm{4}}}}}{{\rm{3}}}\).

Step-by-step solution

Given Data.

\({x^2} + {y^2} + {z^2} \le {a^2}\)

Also, density is proportional to the distance from the z-axis.

Finding Mass of a Ball B.

\(\begin{aligned}\int_{{\rm{ - a}}}^{\rm{a}} {\int_{\rm{0}}^{{\rm{2\pi }}} {\int_{\rm{0}}^{\rm{a}} {{\rm{(Kr)}}} } } \rm rdrd\theta dz &= K\int_{{\rm{ - a}}}^{\rm{a}} {\int_{\rm{0}}^{{\rm{2\pi }}} {\int_{\rm{0}}^{\rm{a}} {{{\rm{r}}^{\rm{2}}}} } } {\rm{drd\theta dz}}\\\rm &= K\int_{{\rm{ - a}}}^{\rm{a}} {\int_{\rm{0}}^{{\rm{2\pi }}} {\left( {\frac{{{{\rm{r}}^{\rm{3}}}}}{{\rm{3}}}} \right)_\rm r = 0^{{\rm{r = a}}}} } {\rm{d\theta dz}}\\\rm &= K\int_{{\rm{ - a}}}^{\rm{a}} {\int_{\rm{0}}^{{\rm{2\pi }}} {\frac{{{{\rm{a}}^{\rm{3}}}}}{{\rm{3}}}} } {\rm{d\theta dz}}\\\rm &= K\int_{{\rm{ - a}}}^{\rm{a}} {\frac{{{\rm{2\pi }}{{\rm{a}}^{\rm{3}}}}}{{\rm{3}}}} {\rm{dz}}\\\rm &= \frac{{{\rm{4\pi }}{{\rm{a}}^{\rm{4}}}}}{{\rm{3}}}\end{aligned}\)

Thus, the Mass of a Ball B is \(\frac{{{\rm{4\pi }}{{\rm{a}}^{\rm{4}}}}}{{\rm{3}}}\).