In this problem you will calculate the area between $f(x) = 6 x^3$ and the $x$-axis over the interval $\lbrack 0, 2 \rbrack$ using a limit of right-endpoint Riemann sums:

$$\mathrm{Area} = \lim_{n \to \infty} \left( \sum_{k=1}^{n} f(x_k) \Delta x \right).$$
Express the following quantities in terms of $n$, the number of rectangles in the Riemann sum, and $k$, the index for the rectangles in the Riemann sum.

1. We start by subdividing $\lbrack 0, 2 \rbrack$ into $n$ equal width subintervals $\lbrack x_0, x_1 \rbrack, \lbrack x_1, x_2 \rbrack, \ldots, \lbrack x_{n-1}, x_{n} \rbrack$ each of width $\Delta x$. Express the width of each subinterval $\Delta x$ in terms of the number of subintervals $n$.
   $\Delta x =$
   
   
2. Find the right endpoints $x_1, x_2, x_3$ of the first, second, and third subintervals $\lbrack x_0, x_1 \rbrack, \lbrack x_1, x_2 \rbrack, \lbrack x_2, x_3 \rbrack$ and express your answers in terms of $n$.
   $x_1, x_2, x_3 =$ (Enter a comma separated list.)
   
   
3. Find a general expression for the right endpoint $x_k$ of the $k$^th subinterval $\lbrack x_{k-1}, x_{k} \rbrack$, where $1 \leq k \leq n$. Express your answer in terms of $k$ and $n$.
   $x_k =$
   
   
4. Find $f(x_k)$ in terms of $k$ and $n$.
   $f(x_k) =$
   
   
5. Find $f(x_k) \Delta x$ in terms of $k$ and $n$.
   $f(x_k) \Delta x =$
   
   
6. Find the value of the right-endpoint Riemann sum in terms of $n$.
   $\displaystyle \sum_{k=1}^{n} f(x_k) \Delta x =$
   
   
7. Find the limit of the right-endpoint Riemann sum.
   $\displaystyle \lim_{n\to\infty} \left( \sum_{k=1}^{n} f(x_k) \Delta x \right) =$