2. Given the function $f(x) = x^3$, do the following to integrate $\int_0^c x^3 dx$.
(a) Find $\Delta x$ in terms of $n$ and the limits of integration (0 and $c$), where $n$ is
any arbitrary natural number.
(b) Find $x_k$ in terms of $k$ and $\Delta x$.
(c) Assume that $x_k^* = x_k$ (RRAM method), and express $f(x_k^*) = f(x_k)$ in
terms of $k$, $c$, and $n$.
(d) Express the following integral as a limit of a Riemann sum involving $k$, $c$,
and $n$ by using definition of the definite integral as a limit.
$\int_0^c x^3 dx$
(e) Express the summation within the limit as a sum in expanded form, and
show that
$\int_0^c x^3 dx = \lim_{n \to \infty} \frac{c^4}{n^4} (1^3 + 2^3 + \dots + n^3)$.
(f) Replace the sum in parenthesis using the rule
$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$, and find the value of the integral in (e) in
terms of $c$.
