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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 $c$.

          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 $c$.

2. Given the function f(x) = x^3, do the following to integrate ∫0^c x^3 dx.
(a) Find Δ x in terms of n and the limits of integration (0 and c), where n is
any arbitrary natural number.
(b) Find xk in terms of k and Δ x.
(c) Assume that xk^* = xk (RRAM method), and express f(xk^*) = f(xk) 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.
∫0^c x^3 dx
(e) Express the summation within the limit as a sum in expanded form, and
show that
∫0^c x^3 dx = limn →∞(c^4)/(n^4) (1^3 + 2^3 + … + n^3).
(f) Replace the sum in parenthesis using the rule
1^3 + 2^3 + … + n^3 = (n^2(n+1)^2)/(4), and find the value of the integral in (e) in
terms of c.

Added by Melody C.

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