Answer
$\frac{11}{3}$
Work Step by Step
Step 1. See figure. Find the intersections between the functions: the intersections are at $(1,2), (4,1)$
Step 2. The enclosed area can be written as the sum of two integrations:
$A=\int_{0}^1 (1+\sqrt x-\frac{x}{4})dx + \int_1^4 (\frac{2}{\sqrt x}-\frac{x}{4})dx =x|_{0}^1+\frac{2}{3}x^{3/2}|_{0}^1-\frac{1}{8}x^2|_{0}^1+4\sqrt x|_1^4-\frac{1}{8}x^2 |_1^4=1+\frac{2}{3}-\frac{1}{8}+4\sqrt 4-4\sqrt 1-\frac{1}{8}(4)^2+\frac{1}{8}(1)^2=\frac{11}{3}$
