$\text{The value of }\underset{n\to \infty }{lim} \frac{1}{n^4}\left(\sum _{r=1}^{2n}\right(3n{r}^2+2n^{2}r\left)\right) is equal to$

$\begin{array}{l}\underset{n\to \infty }{ lim} \frac{1}{n^4}\sum _{r=1}^{2n}\left(3n{r}^2+2n^{2}r\right)\text{}=\underset{n\to \infty }{lim} \frac{1}{n^4}{n}^3 \sum _{r=1}^{2n}\left(3{\left(\frac{r}{n}\right)}^2+2\left(\frac{r}{n}\right)\right)\\ ={\int }_0^2\left(3x^2+2x\right)dx\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}={\left(x^3+x^2\right)}_0^2\text{}\\ =8+4=12\\ \end{array}$