Chapter 8.6, Problem 16E

To determine

To evaluate: The double integral ${\displaystyle \underset{R}{\iint }f\left(x,y\right)}dA$ where $f\left(x,y\right)=x^2{y}^2$, R is bounded by the lines $x=0$, $x=1$, $y=x^2$ and $y=x^3$.