[m]f`(x)=\frac{2}{3}\cdot \frac{3}{2}\cdot x^{\frac{1}{2}}-\frac{1}{2}\cdot \frac{1}{2}\cdot x^{-\frac{1}{2}}[/m]
[m]f`(x)=\sqrt{x}-\frac{1}{4\sqrt{x}}[/m]
[m]1+(f`(x))^2=1+(\sqrt{x}-\frac{1}{4\sqrt{x}})^2=1+x-\frac{1}{2}+\frac{1}{16x}=x+\frac{1}{2}+\frac{1}{16x}=(\sqrt{x}+\frac{1}{4\sqrt{x}})^2[/m]
[m]\sqrt{1+(f`(x))^2}=\sqrt{x}+\frac{1}{4\sqrt{x}}[/m]
[m]s= ∫_{1} ^{4}(\sqrt{x}+\frac{1}{4\sqrt{x}})dx=(\frac{x^{ \frac{3}{2}}}{ \frac{3}{2}}+\frac{1}{4}\cdot 2\sqrt{x})_{1} ^{4}=[/m]
[m]=(\frac{3}{2}\sqrt{x^3}+\frac{1}{2}\sqrt{x})|_{1} ^{4}=(\frac{3}{2}\sqrt{4^3}+\frac{1}{2}\sqrt{4})-(\frac{3}{2}\sqrt{1^3}+\frac{1}{2}\sqrt{1})=(\frac{3}{2}\cdot 8+1)-(\frac{3}{2}+\frac{1}{2})=13-2=11[/m]