Lecture 20: Taylor's Theorem and the Definition of Riemann Sums | Summary and Q&A

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June 21, 2022
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Lecture 20: Taylor's Theorem and the Definition of Riemann Sums

TL;DR

Riemann integration assigns a number to a function that represents the area under its curve, but it is not an exact measurement.

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Key Insights

  • 🍹 Riemann integration is a method for approximating the area under a curve by dividing the interval into smaller subintervals and computing the sum of the function values multiplied by the width of each subinterval.
  • 🍹 The Riemann sum assigns a number to a partition and tag of an interval, representing an approximate area under the curve.
  • 🍹 The Riemann integral is defined as the limit of the Riemann sums as the norm of the partition approaches zero.

Transcript

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Questions & Answers

Q: What is Riemann integration?

Riemann integration is a method for approximating the area under a curve by dividing the interval into smaller subintervals and computing the sum of the function values multiplied by the width of each subinterval.

Q: How is a partition created?

A partition is created by dividing the interval into smaller subintervals, with each subinterval having its left and right endpoints.

Q: What is the Riemann sum?

The Riemann sum is a number obtained by multiplying the function value at a tag point within each subinterval by the width of the subinterval, and summing these products.

Q: How is the Riemann integral defined?

The Riemann integral is defined as the limit of the Riemann sums as the norm of the partition approaches zero.

Summary & Key Takeaways

  • Riemann integration is a method for approximating the area under a curve using a series of sums.

  • A partition of the interval is created by dividing it into smaller subintervals.

  • Each subinterval is associated with a tag, and the Riemann sum is computed by multiplying the function value at the tag by the width of the subinterval.

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