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

Step 1 ：The formula for the right Riemann sum is given by: \[R_{N}=\frac{b-a}{N} \sum_{i=1}^{N} f\left(a+i \frac{b-a}{N}\right)\] 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: \[R_{N}=\frac{5-(-1)}{N} \sum_{i=1}^{N} \left(12\left(-1+i \frac{5-(-1)}{N}\right)\right)^2\]

Step 4 ：Simplify this to: \[R_{N}=\frac{6}{N} \sum_{i=1}^{N} \left(12\left(-1+\frac{6i}{N}\right)\right)^2\]

Step 5 ：Simplify further to: \[R_{N}=\frac{6}{N} \sum_{i=1}^{N} \left(144\left(-1+\frac{6i}{N}\right)^2\right)\]

Step 6 ：Simplify further to: \[R_{N}=864 \sum_{i=1}^{N} \left(-1+\frac{6i}{N}\right)^2\]

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 \(R_{N}\): \[\lim _{N \rightarrow \infty} R_{N}=\lim _{N \rightarrow \infty} 864 \sum_{i=1}^{N} \left(-1+\frac{6i}{N}\right)^2\]

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: \[\left[\frac{(12x)^3}{3}\right]_{-1}^{5} = \frac{(12*5)^3}{3} - \frac{(12*-1)^3}{3} = 43200 - (-1728) = 44928\]

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