Find a formula for $R_N$ and compute the area under the graph of $f(x)=(12x{)}^2$ over $[-1,5]$ as a limit.
(Give your answer as a whole or exact number.)
$$\underset{N\to \mathrm{\infty }}{lim}{R}_N=$$

Step 1 ：The formula for the right Riemann sum is given by: $$R_N=\frac{b-a}{N}\sum _{i=1}^{N}f(a+i\frac{b-a}{N})$$ where $[a,b]$ is the interval over which we are integrating, $N$ is the number of rectangles we are using to approximate the area, and $f(x)$ is the function we are integrating.

Step 2 ：In this case, $a=-1$, $b=5$, and $f(x)=(12x{)}^2$.

Step 3 ：So, we have: $$R_N=\frac{5-(-1)}{N}\sum _{i=1}^N{\left(12(-1+i\frac{5-(-1)}{N})\right)}^2$$

Step 4 ：Simplify this to: $$R_N=\frac{6}{N}\sum _{i=1}^N{\left(12(-1+\frac{6i}{N})\right)}^2$$

Step 5 ：Simplify further to: $$R_N=\frac{6}{N}\sum _{i=1}^N\left(144{(-1+\frac{6i}{N})}^2\right)$$

Step 6 ：Simplify further to: $$R_N=864\sum _{i=1}^N{(-1+\frac{6i}{N})}^2$$

Step 7 ：Now, we want to compute the area under the graph of $f(x)=(12x{)}^2$ over $[-1,5]$ as a limit. This is given by the limit as $N$ approaches infinity of $R_N$: $$\underset{N\to \mathrm{\infty }}{lim}{R}_N=\underset{N\to \mathrm{\infty }}{lim}864\sum _{i=1}^N{(-1+\frac{6i}{N})}^2$$

Step 8 ：This is a limit of a Riemann sum, which is the definition of an integral. So, this is equal to the integral of $f(x)$ from $-1$ to $5$: $${\int }_{-1}^5(12x{)}^{2}dx$$

Step 9 ：This integral can be computed as: $${\left[\frac{(12x{)}^3}{3}\right]}_{-1}^5=\frac{(12\ast 5{)}^3}{3}-\frac{(12\ast -1{)}^3}{3}=43200-(-1728)=44928$$

Step 10 ：$\boxed{{\displaystyle 44928}}$ is the area under the graph of $f(x)=(12x{)}^2$ over $[-1,5]$.