(Solved): Consider the following definite integral. 011(11xx2)dx a. Write the midpoint Riemann sum i ...

Consider the following definite integral. ${?}_0^{11}\left(11x?x^2\right)dx$ a. Write the midpoint Riemann sum in sigma notation for an arbitrary value of $n$. b. Evaluate the sum using a calculator with $n=20,50$, and 100 . Use these values to estimate the value of the integral. a. Choose the correct Riemann sum below. A. ${?}_{k=1}^n(k)(n?1)\left(\frac{1331}{4n^3}\right)$ C. ${?}_{k=1}^n(k?1)(n+1?k)\left(\frac{1331}{n^3}\right)$ E. ${?}_{k=1}^n(2k?1)(2n+1?2k)\left(\frac{1331}{n^3}\right)$ B. ${?}_{k=1}^n(2k?1)(2n+1?2k)\left(\frac{1331}{4n^3}\right)$ D. ${?}_{k=1}^n(k)(n?1)\left(\frac{1331}{n^3}\right)$ F. ${?}_{k=1}^n(k?1)(n+1?k)\left(\frac{1331}{4n^3}\right)$ b. Evaluate the sum using a calculator, with $\mathrm{n}=20$. The sum with $n=20$ is (Do not round until the final answer. Then round to four decimal places as needed.)