Midpoint Riemann sum approximation

[Leo Liu]

Homework Statement

This equation is found on PAGE 7, https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-3-the-definite-integral-and-its-applications/exam-3/materials-for-exam-3/MIT18_01SCF10_exam3sol.pdf

Relevant Equations

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Can someone please explain why the formula for midpoint approximation looks like the equation above instead of something like
$$M_n=(f(\frac{x_0+x_1}2)+f(\frac{x_1+x_2}2)+\cdots+f(\frac{x_{n-1}+x_n}2))\frac{b-a}n$$?
Thanks in advance!

 

[FactChecker]

In order to have enough partitions for the Simpson's Rule approximation, they have twice as many partitions as they are using for the midpoint approximation. So each of the midpoints that you are calculating by using an average is actually an exact partition point in their Simpson's Rule partitioning.