Problem $12.$ A Petri dish $in$ the shape $of$ a rectangle $4cmby6cm$ contains a growing bacterial colony. The

density $of$ bacteria $at$ a point $(x,y),$ measured $in$ millions $of$ cells per $c{m}^2,is$ modeled $by$

$(x,y)=5+2sin(\frac{x}{4})cos(\frac{y}{6}),$

where $0x4$ and $0y6\{$:$=\frac{1}{\text{A}\text{r}\text{e}\text{a}}{}_{D}f(x,y)dA.)$