Answer
$\frac{5}{2}$
Work Step by Step
An equation of the line through $(0,0)$ and $(3,1)$ is$ y=\frac{1}{3}x$;through $(0,0)$ and $(1,2)$ is $y=2x$;through $(3,1)$ and $(1,2)$ is $ y=-\frac{1}{2}x+\frac{5}{2}.$
$A=\int_{0}^{1}(2x-\frac{1}{3}x)dx$$+$ $\int_{1}^{3} [(-\frac{1}{2}x+\frac{5}{2})-\frac{1}{3}x]dx$
$=\int_{0}^{1}\frac{5}{3}xdx$ $+$ $\int_{1}^{3}(-\frac{5}{6}+\frac{5}{2})dx$
$=[\frac{5}{6}x^2]_{0}^{1}$ $+$ $[-\frac{5}{12}x^2 +\frac{5}{2}x]_{1}^{3}$
$=\frac{5}{6}+(-\frac{15}{4}+\frac{15}{2})-(-\frac{5}{12}+\frac{5}{2})$
$=\frac{5}{2}$
