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

TL;DR
Riemann integration assigns a number to a function that represents the area under its curve, but it is not an exact measurement.
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
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Riemann integration is a method for approximating the area under a curve using a series of sums.
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A partition of the interval is created by dividing it into smaller subintervals.
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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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