Numeric Integration, Part 6

A function f(x) has a graph which is decreasing and concave-down over the interval 0 < x < 2. (You should make a rough sketch of the graph on scratch paper to give yourself a visual reference for this problem.)
We are given that the results of approximating the definite integral of f(x) over the interval [0, 2] using left-hand sums, right-hand sums, trapezoidal sums, and midpoint sums with n = 30 subintervals are the numbers 8.1311, 7.9977, 7.8644, and 8.0011. The only trouble is that these numbers have been scrambled and we aren't told which number comes from which approximation method. Match each number with the approximation method that must have produced it. Explain how you know which is which.

Given just the numbers listed in boldface in step 1, find the highest accuracy approximation that you can for the value of the integral of f(x) over the interval [0, 2] using only a simple "averaging" method.

For a certain integral, suppose the error in the LHS approximation using n = 20 subintervals is 0.002148. Approximately what would you expect the error to be if you computed the LHS approximation with n = 40 subintervals?

In the previous step, suppose the approximation method was really MID, not LHS. How would your answer change? Answer the same question for TRAP and for Simpson's Rule.