MvCalc17

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I=\int _{0}^{1}\int _{{-{\sqrt  {y}}}}^{{{\sqrt  {y}}}}dxdy+\int _{1}^{9}\int _{{{\frac  {y-3}{2}}}}^{{{\sqrt  {y}}}}dxdy\,

Now integrating the two functions separately,

\int _{0}^{1}[\int _{{-{\sqrt  {y}}}}^{{{\sqrt  {y}}}}dx]dy=\int _{0}^{1}[x]_{{-{\sqrt  {y}}}}^{{{\sqrt  {y}}}}dy\,

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

\int _{1}^{9}[\int _{{{\frac  {y-3}{2}}}}^{{{\sqrt  {y}}}}dx]dy=\int _{1}^{9}[x]_{{{\frac  {y-3}{2}}}}^{{{\sqrt  {y}}}}dy\,

=\int _{1}^{9}[{\sqrt  {y}}-{\frac  {y-3}{2}}]dy=[y^{{{\frac  {3}{2}}}}-{\frac  {1}{2}}({\frac  {y^{2}}{2}}-3y)]_{1}^{9}\,

=[9^{{{\frac  {3}{2}}}}-{\frac  {1}{2}}({\frac  {81}{2}}-27)]-[1-{\frac  {1}{2}}(1-3)]\,

=[27-{\frac  {1}{2}}{\frac  {27}{2}}]-[1+1]={\frac  {81}{4}}-2={\frac  {73}{4}}\, --(2)

Adding (1) and (2), we get I={\frac  {73}{4}}+1={\frac  {77}{4}}\,

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