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:

Express the following quantities in terms of

- We start by subdividing
\lbrack 0, 2 \rbrack inton 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 subintervalsn .

\Delta x =

- 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 ofn .

x_1, x_2, x_3 = (Enter a comma separated list.)

- Find a general expression for the right endpoint
x_k of thek ^{th}subinterval\lbrack x_{k-1}, x_{k} \rbrack , where1 \leq k \leq n . Express your answer in terms ofk andn .

x_k =

- Find
f(x_k) in terms ofk andn .

f(x_k) =

- Find
f(x_k) \Delta x in terms ofk andn .

f(x_k) \Delta x =

- Find the value of the right-endpoint Riemann sum in terms of
n .

\displaystyle \sum_{k=1}^{n} f(x_k) \Delta x =

- 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) =

You can earn partial credit on this problem.