Chapter 5 - Integration - 5.3 Fundamental Theorem of Calculus - 5.3 Exercises - Page 374: 51

$$A = \frac{{14}}{3}$$

$$\eqalign{ & \left( i \right){\text{The net area of the region is given by:}} \cr & A = \int_1^4 {{x^{1/2}}} dx \cr & {\text{Integrating}} \cr & A = \left[ {\frac{{{x^{3/2}}}}{{3/2}}} \right]_1^4 \cr & A = \frac{2}{3}\left[ {{x^{3/2}}} \right]_1^4 \cr & {\text{Evaluating}} \cr & A = \frac{2}{3}\left( {{4^{3/2}} - {1^{3/2}}} \right) \cr & A = \frac{2}{3}\left( {8 - 1} \right) \cr & A = \frac{{14}}{3} \cr} $$