Answer
$\frac {41}{24}$
Work Step by Step
Since, $D=(x,y) | 0\leq y\leq 1, y^{2}\leq x \leq y+2$
$\int\int_{D} xydA=\int_{0}^{1}\int_{y^{2}} ^{y+2}xydxdy$
$=\int_{0}^{1}[\frac{x^{2}}{y}]_{y^{2}} ^{y+2}dxdy$
$=(\frac {y^{4}}{8}+\frac{2}{3}y^{3}+y^{2}-\frac{y^{6}}{12}]_{0} ^{1}$
$=\frac{1}{8}+\frac{2}{3}+1-\frac{1}{12}$
$=\frac {41}{24}$
