# Thoughts on Numerical Integration (Part 21): Simpson’s rule and global rate of convergence

Numerical integration is a standard topic in first-semester calculus. From time to time, I have received questions from students on various aspects of this topic, including:
• Why is numerical integration necessary in the first place?
• Where do these formulas come from (especially Simpson’s Rule)?
• How can I do all of these formulas quickly?
• Is there a reason why the Midpoint Rule is better than the Trapezoid Rule?
• Is there a reason why both the Midpoint Rule and the Trapezoid Rule converge quadratically?
• Is there a reason why Simpson’s Rule converges like the fourth power of the number of subintervals?
In this series, I hope to answer these questions. While these are standard questions in a introductory college course in numerical analysis, and full and rigorous proofs can be found on Wikipedia and Mathworld, I will approach these questions from the point of view of a bright student who is currently enrolled in calculus and hasn’t yet taken real analysis or numerical analysis.
In this previous post in this series, we showed that the Simpson’s Rule approximation of $\displaystyle \int_{x_i}^{x_i+2h} x^k \, dx$ has an error of

$-\displaystyle \frac{k(k-1)(k-2)(k-3)}{90} x_i^{k-4} h^5 + O(h^6)$.

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