- #1
imull
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I am really having trouble understanding how upper and lower sums are calculated. In the equations Ʃf(Mi)Δx and Ʃf(mi)Δx, what do Mi and mi represent?
Upper & Lower Sums: Calculating & Understanding Mi & mi
- Context: Undergrad
- Thread starter imull
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Discussion Overview
The discussion revolves around the calculation and understanding of upper and lower sums in the context of Riemann sums, specifically focusing on the notation used for Mi and mi. Participants explore the definitions and interpretations of these terms within the framework of calculus.
Discussion Character
- Technical explanation
- Debate/contested
Main Points Raised
- One participant expresses confusion about the calculation of upper and lower sums and the meaning of Mi and mi.
- Another participant suggests that visual aids, such as figures showing rectangles representing upper and lower sums, can help clarify the concepts.
- There is a contention regarding the interpretation of Mi and mi, with one participant asserting that Mi refers to x values where f(Mi) gives the heights of the upper rectangles, while mi refers similarly for the lower rectangles.
- A later reply questions the notation used, suggesting it may be unusual and implying a possible error in the original text from which the notation was copied.
- One participant acknowledges a misreading of the question, indicating a misunderstanding of the notation.
- The original poster confirms that the notation was copied directly from their text, indicating no error on their part.
Areas of Agreement / Disagreement
Participants exhibit disagreement regarding the interpretation of the notation Mi and mi, with some asserting one definition while others challenge it. The discussion remains unresolved as different interpretations are presented without consensus.
Contextual Notes
The discussion highlights potential confusion arising from notation that may not be standard across different texts, leading to varying interpretations of the terms Mi and mi.
- #2
jbunniii
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It's easiest to understand if you look at a picture. In the figure below (shamelessly borrowed from elsewhere via Google Images), the pink rectangles represent the upper sums and the green rectangles represent the lower sums.
In other words, the ##M_i## are the heights of the taller (pink + green) rectangles, and the ##m_i## are the heights of the shorter (green only) rectangles.
How are the heights determined? Quite simple. Each tall rectangle is exactly as high as the maximum value of the function within the width of the rectangle. Each short rectangle is exactly as high as the minimum value of the function.
In other words, the ##M_i## are the heights of the taller (pink + green) rectangles, and the ##m_i## are the heights of the shorter (green only) rectangles.
How are the heights determined? Quite simple. Each tall rectangle is exactly as high as the maximum value of the function within the width of the rectangle. Each short rectangle is exactly as high as the minimum value of the function.
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- #3
pwsnafu
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jbunniii said:
No. ##M_i## are the x values such that ##f(M_i)## are the heights of the taller rectangles. Similarly for ##m_i##.
- #4
LCKurtz
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You're correct of course, but that is such unusual notation it makes you wonder if the OP copied the expression correctly from his text.
- #5
jbunniii
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Oh, weird. You're right. I misread the question to say what it would have said in most books.pwsnafu said:
- #6
imull
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I did copy them from my text. I wasn't sure how else to go about this.
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