For the function given below, find a formula for the Riemann sum obtained by dividing the interval $[a,b]$ into $n$ equal subintervals and using the right-hand endpoint for each $c_k$. Then take a limit of this sum as $n\to \mathrm{\infty }$ to calculate the area under the curve over $[a,b]$.
$f(x)=3x$ over the interval $[2,6]$
Find a formula for the Riemann sum.
$$S_n=$$

Step 1 ：Given the function $f(x)=3x$ over the interval $[2,6]$, we are asked to find a formula for the Riemann sum obtained by dividing the interval into $n$ equal subintervals and using the right-hand endpoint for each $c_k$.

Step 2 ：The formula for the Riemann sum is given by $S_n=\sum _{k=1}^{n}f(a+k\cdot \mathrm{\Delta }x)\cdot \mathrm{\Delta }x$, where $\mathrm{\Delta }x=\frac{b-a}{n}$ is the width of each subinterval, and $f(a+k\cdot \mathrm{\Delta }x)$ is the height of the $k$-th rectangle.

Step 3 ：In this case, $a=2$, $b=6$, and $f(x)=3x$. So, $\mathrm{\Delta }x=\frac{6-2}{n}=\frac{4}{n}$, and $f(a+k\cdot \mathrm{\Delta }x)=3(a+k\cdot \mathrm{\Delta }x)=3(2+k\cdot \frac{4}{n})=\frac{12k}{n}+6$.

Step 4 ：Substituting these values into the formula for the Riemann sum, we get $S_n=\sum _{k=1}^n(\frac{12k}{n}+6)\cdot \frac{4}{n}=24+48(\frac{n^2}{2}+\frac{n}{2})/n^2$.

Step 5 ：Now, we need to take the limit of this sum as $n\to \mathrm{\infty }$ to calculate the area under the curve over $[2,6]$.

Step 6 ：As $n\to \mathrm{\infty }$, the term $\frac{n^2}{2}+\frac{n}{2}$ in the Riemann sum becomes very large, and the term $24+48(\frac{n^2}{2}+\frac{n}{2})/n^2$ approaches 48.

Step 7 ：Final Answer: The area under the curve $f(x)=3x$ over the interval $[2,6]$ is $\boxed{{\displaystyle 48}}$.