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DeMath $\displaystyle {L_y} = \int\limits_a^b {\sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} dx} = \int\limits_1^3 {\sqrt {1 + {{\left( {\frac{d}
{{dx}}2{x^{3/2}}} \right)}^2}} dx} = \int\limits_1^3 {\sqrt {1 + {{\left( {3{x^{1/2}}} \right)}^2}} dx} =$
$\displaystyle = \int\limits_1^3 {\sqrt {1 + 9x} dx} = \frac{1}{9}\int\limits_1^3 {{{\left( {1 + 9x} \right)}^{1/2}}d\left( {1 + 9x} \right)} = \left. {\frac{1}
{9}\left( {\frac{2}{3}{{\left( {1 + 9x} \right)}^{3/2}}} \right)} \right|_1^3 =$
$\displaystyle = \left. {\frac{2}{{27}}{{\left( {1 + 9x} \right)}^{3/2}}} \right|_1^3 = \frac{2}{{27}}\left( {\sqrt {{{28}^3}} - \sqrt {{{10}^3}} } \right) = \frac{2}{{27}}\left( {\sqrt {{{28}^2} \cdot 28} - \sqrt {{{10}^3}} } \right) =$
$\displaystyle = \frac{2}{{27}}\left( {\sqrt {{{28}^2} \cdot {2^2} \cdot 7} - \sqrt {{{10}^2} \cdot 10} } \right) = \frac{2}{{27}}\left( {56\sqrt 7 - 10\sqrt {10} } \right) =$
$\displaystyle = \frac{4}{{27}}\left( {28\sqrt 7 - 5\sqrt {10} } \right) \approx 8.632540504$