Chapter 14.2, Problem 2E

To determine

To calculate: The approximate value of the integral $\frac{1}{2}{\displaystyle {\int }_0^4{\displaystyle {\int }_0^2\left(x^{2}y\right)dydx}}$ by dividing the rectangle R with vertices $\left(0,0\right),\left(4,0\right)\left(4,2\right)\text{ and }\left(0,2\right)$ into eight equal squares and finding the sum ${\displaystyle \sum _{i=1}^8\left(\left(x_i{}^2\right)y_i\right)\Delta A_i}$, where $\left(x_i,y_i\right)$ is the center of the $i^{th}$ square and compare this value with the value of the integrated integral.