MvCalc18

From Example Problems
Jump to: navigation, search

\int _{{0}}^{{2a}}[\int _{{{\frac  {y^{2}}{4a}}}}^{{3a-y}}(x^{2}+y^{2})dx]dy\,

=\int _{{0}}^{{2a}}[{\frac  {x^{3}}{3}}+y^{2}x]_{{x={\frac  {y^{2}}{4a}}}}^{{3a-y}}dy\,

=\int _{{0}}^{{2a}}[{\frac  {1}{3}}(3a-y)^{3}+(3a-y)y^{2}]-[{\frac  {y^{6}}{192}}+{\frac  {y^{4}}{4a}}]dy\,

=\int _{{0}}^{{2a}}[9a^{3}-3a^{2}y+3ay^{2}-{\frac  {y^{3}}{3}}-{\frac  {y^{6}}{192}}-{\frac  {y^{4}}{4a}}]dy\,

=[9a^{3}y-{\frac  {3}{2}}a^{2}y^{2}+ay^{3}-{\frac  {y^{4}}{12}}-{\frac  {y^{7}}{1344}}-{\frac  {y^{5}}{20a}}]_{{0}}^{{2a}}\,

=9a^{3}(2a)-6a^{4}+8a^{4}-{\frac  {4}{3}}a^{4}-{\frac  {2}{21}}a^{7}-{\frac  {8}{5}}a^{4}\,

=20a^{4}-{\frac  {4}{3}}a^{4}-{\frac  {8}{5}}a^{4}-{\frac  {2}{21}}a^{7}\,

={\frac  {256}{15}}a^{4}-{\frac  {2}{21}}a^{7}=a^{4}[{\frac  {256}{15}}-{\frac  {2}{21}}{a^{3}}]\,


Main Page