My brother had this question for a homework assignment in calculus. The question speaks for itself and seems easy enough to solve. However, it seems that the midpoint approximation here is incorrect. My understanding is that midpoint approximation is the average of the lower and upper approximations, but the answers presented in these examples do not hold true with that.
I chose these examples specifically because they only differ in two values (both of which are the larger in the second example), and have the same solution for the midpoint approximation.
What I am asking for is either an explanation of what we've been doing wrong or, in the case that the system is wrong, a method to find its solution rather than our own. Thanks all
Best Answer
The average of the left and right endpoint estimates, sometimes called $L_n$ and $R_n$ respectively, is actually the same as the trapezoid estimate, called sometimes $T_n$, which is based on drawing trapezoidal slabs based on the values of the function at the two endpoints.
One would not expect the midpoint rule $M_n$ to agree in general with any of the above, since it uses sample values of the function at the midpoints of the subintervals, and those values are independent (in general) of the values of the function at the endpoints of the subintervals.
ADDED: I see now what they did in the "midpoint rule" calculation. For it they actually used three intervals of length $2$, so that the values at the midpoints are actually on the table of values given, and are (from left to right) the numbers $8,38,56$ with a sum of $102$, which must be multiplied by the subinterval length of $2$ to get the answer of $204$ as stated in the printout. (By the way I had to zoom in a bit to see that.)