[tex]S=\int\limits^4_1 {\sqrt{x} } \, dx = \int\limits^4_1 {x^{\frac{1}{2} }} \, dx = \frac{2\sqrt{x}\; |x|}{3} |^{4} _{1} =\frac{2\sqrt{4}\; |4|}{3}-\frac{2\sqrt{1}\; |1|}{3}=\frac{14}{3} .[/tex]