In this problem you will calculate the area between $f(x) = 5x^2$ and the $x$-axis over the interval $[0, 3]$ using a limit of right-endpoint Riemann sums:
$\text{Area} = \lim_{n \to \infty} \sum_{k=1}^{n} f(x_k) \Delta x$
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.
a. We start by subdividing $[0, 3]$ into $n$ equal width subintervals $[x_0, x_1], [x_1, x_2], \dots, [x_{n-1}, x_n]$ each of width $\Delta x$. Express the width of each
subinterval $\Delta x$ in terms of the number of subintervals $n$.
$\Delta x =
$
b. Find the right endpoints $x_1, x_2, x_3$ of the first, second, and third subintervals $[x_0, x_1], [x_1, x_2], [x_2, x_3]$ and express your answers in terms of $n$.
(Enter a comma separated list.)
$x_1, x_2, x_3 =
$
c. Find a general expression for the right endpoint $x_k$ of the $k^{th}$ subinterval $[x_{k-1}, x_k]$, where $1 \le k \le n$. Express your answer in terms of $k$ and $n$.
$x_k =
$
d. Find $f(x_k)$ in terms of $k$ and $n$.
$f(x_k) =
$
e. Find $f(x_k) \Delta x$ in terms of $k$ and $n$.
$f(x_k) \Delta x =
$
f. Find the value of the right-endpoint Riemann sum in terms of $n$.
$\sum_{k=1}^{n} f(x_k) \Delta x =
$
g. Find the limit of the right-endpoint Riemann sum.
$\lim_{n \to \infty} \left( \sum_{k=1}^{n} f(x_k) \Delta x \right) =
$
