Answer
$$A=ln2$$
Work Step by Step
From the Graph, we can see that from 0 through 1, the top curve is y=x and the bottom curve is y=$\frac{1}{4}$. From 1 through 2 though, the top curve changes to y=$\frac{1}{x}$
$$A=\int{^1_0}(x-\frac{1}{4}x) dx + \int{^2_1}(\frac{1}{x}-\frac{1}{4}x) dx$$
$$A=\frac{1}{2}x^{2}-\frac{1}{8}x^{2}|{^1_0} + ln|x|-\frac{1}{8}x^{2}|{^2_1}$$
$$A=(\frac{4}{8}-\frac{1}{8}) + ((ln2-\frac{1}{2})-(0-\frac{1}{8}))$$
$$A=\frac{3}{8} + (ln2 - \frac{4}{8}+\frac{1}{8})$$
$$A=\frac{3}{8}-\frac{3}{8}+ln2$$
$$A=ln2$$
