Midpoint Riemann sum approximation

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SUMMARY

The discussion centers on the formula for the midpoint Riemann sum approximation, specifically addressing its structure compared to other approximation methods. The midpoint approximation is defined as \(M_n = (f(\frac{x_0+x_1}{2}) + f(\frac{x_1+x_2}{2}) + \cdots + f(\frac{x_{n-1}+x_n}{2})) \cdot \frac{b-a}{n}\). It is clarified that Simpson's Rule requires twice as many partitions as the midpoint approximation, linking the midpoints directly to the partition points used in Simpson's Rule.

PREREQUISITES
  • Understanding of Riemann sums
  • Familiarity with Simpson's Rule
  • Basic calculus concepts, including integration
  • Knowledge of partitioning in numerical methods
NEXT STEPS
  • Study the derivation of the midpoint Riemann sum formula
  • Learn about Simpson's Rule and its application in numerical integration
  • Explore the differences between various numerical approximation methods
  • Investigate the implications of partition sizes on approximation accuracy
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Students and educators in mathematics, particularly those focusing on calculus and numerical methods, as well as professionals involved in mathematical modeling and computational analysis.

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
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1612402856532.png

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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