What does the index of summation in sigma notation represent?

Calculate the midpoint Riemann sum for f(x) = x^2 over the interval [0, 2] with 2 subintervals.

Express the following limit of Riemann sums as a definite integral by identifying the function:

$\underset{\Delta \to 0}{\lim }\sum _{k=1}^n\left(x_k^{\ast }{)}^2{e}^{x_k^{\ast }}\Delta x_k\text{ on }\right[0,1]$

Apply the Midpoint Rule to approximate ${\int }_2^{10}(x^2-3x)dx$ using $n=8$ subintervals. Then, calculate the absolute error. The exact value of the integral is $\frac{560}{3}$.

Let $h\left(x\right)$ be a function with $\mid h^{\left(4\right)}\left(x\right)\mid \le 3$ on $[2,8]$. Simpson's Rule is used with $n=10$ subintervals to approximate ${\int }_2^{8}h\left(x\right)dx$. A student claims the absolute error $E_S$ satisfies $E_S\le \frac{(\Delta x{)}^5}{8}$. Is this claim correct?