MvCalc63

From Example Problems
Jump to: navigation, search

Now,V={\frac  {1}{8}}{\frac  {4}{3}}\pi a^{3}={\frac  {1}{6}}\pi a^{3}\,

Here,z_{1}={\frac  {\iiint zdV}{\iiint dV}}\,

{\frac  {1}{6}}\pi a^{3}z_{1}=\iiint zdV\,

=\int _{{0}}^{{{\frac  {\pi }{2}}}}\int _{{0}}^{{{\frac  {\pi }{2}}}}\int _{0}^{a}(r\cos \phi )r^{2}\sin \phi drd\theta d\phi \,

={\frac  {a^{4}}{4}}[{\frac  {\sin ^{2}\phi }{2}}]_{{0}}^{{{\frac  {\pi }{2}}}}{\frac  {\pi }{2}}\,

={\frac  {1}{16}}\pi a^{4}\,

Therefore,z_{1}={\frac  {{\frac  {1}{16}}\pi a^{4}}{{\frac  {1}{16}}\pi a^{3}}}={\frac  {3}{8}}a\,

By symmetry,we have x_{1}=y_{1}=z_{1}={\frac  {3}{8}}a\,

Main Page