MvCalc5

From Example Problems
Jump to: navigation, search

I=\int _{0}^{a}[\int _{0}^{b}(x^{2}+y^{2})dx]dy\,

\int _{0}^{b}(x^{2}+y^{2})dx=\int _{0}^{b}x^{2}dx+\int _{0}^{b}y^{2}dx\,

=[{\frac  {x^{3}}{3}}]_{{x=0}}^{{b}}+[y^{2}x]_{{x=0}}^{{b}}\,

={\frac  {b^{3}}{3}}+by^{2}\,

Therefore,I=\int _{0}^{a}({\frac  {b^{3}}{3}}+by^{2})dy\,

=\int _{0}^{a}{\frac  {b^{3}}{3}}dy+\int _{0}^{a}by^{2}dy\,

=[{\frac  {b^{3}y}{3}}]_{0}^{a}+[{\frac  {by^{3}}{3}}]_{0}^{a}\,

={\frac  {ab^{3}}{3}}+{\frac  {a^{3}b}{3}}\,

={\frac  {ab}{3}}(a^{2}+b^{2})\,

Main Page