Midpoint Riemann sum approximation

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The midpoint Riemann sum approximation formula is structured to utilize the average of function values at midpoints of intervals, rather than endpoints. This method ensures that the approximation captures the behavior of the function more accurately within each subinterval. In contrast, Simpson's Rule requires twice as many partitions, as it combines both midpoint and endpoint evaluations for greater accuracy. The midpoints used in the midpoint approximation serve as exact partition points in the context of Simpson's Rule. Understanding these relationships clarifies the rationale behind the midpoint approximation formula.
Leo Liu
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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!
 
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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.
 
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