Geometric meaning of reversing the limits of an integral

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Discussion Overview

The discussion revolves around the geometric interpretation of reversing the limits of an integral, exploring how this operation affects the representation of area and orientation. Participants examine the implications of this reversal in the context of Riemann sums and the broader framework of calculus.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants question how reversing the limits of an integral can be understood geometrically, suggesting that the area itself remains unchanged despite the limits being swapped.
  • Others argue that reversing the limits results in a 'negative area' when viewed in the context of direction along the x-axis, linking this to the concept of orientation in geometry.
  • A participant proposes that reversing limits could be seen as repositioning the endpoints of the area under consideration, maintaining the same area but altering the function's representation.
  • There is a discussion about whether Riemann sums are defined based on direction, with some asserting that orientation does not affect the area calculation, while others contend that orientation is inherent in the definitions used.
  • Some participants express concern that ignoring orientation could complicate calculus, particularly when calculating areas under curves that yield zero when oriented.
  • A distinction is made between different interpretations of calculus, with some participants seeking clarification on the difference between calculus as a framework for calculations and calculus as analysis.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the geometric implications of reversing integral limits. There are competing views on the significance of orientation and its impact on area calculations, as well as differing interpretations of the role of Riemann sums in this context.

Contextual Notes

Limitations include varying definitions of Riemann sums and the ambiguity surrounding the interpretation of orientation in calculus. The discussion reflects a range of assumptions about the nature of area and its relationship to integral limits.

  • #31
In a not-so-dissimilar vein -- Physician says to man "you should stop smoking; you'll live longer". Man replies "but Doc, I'm 95 years old already". Doc says "yes, and if you hadn't smoked, you'd be much older".
 
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  • #32
Here's an example where the sign makes a difference.
If f(x) represents a conservative force, where x is a displacement,
then the signed area under the curve represents the signed-work done by the force.
I treat the points on the curve (x,f(x)) as a sequence of "states" in F-x space (as one would do in a Pressure-Volume diagram in thermodynamics).
Switching the limits corresponds to traversing those states in reverse,
yielding the opposite-sign of the work-done by that force.
 
  • #33
Calculus allows for continuously changing object, standard Algebra does, not, it lives in a discrete realm. You cannot speak, e.g. of convergence or infinite sums within Algebra.
 
  • #34
Also remember integrals may tepresent things other than areas. Consider velocity in x axis, time in other . Then integral represents total distance. If x-axis changes sign then this is just, as Fresh said, a change of orientation. as in Physics. Distance is measured in negative sense.
 
  • #35
WWGD said:
Calculus allows for continuously changing object, standard Algebra does, not, it lives in a discrete realm. You cannot speak, e.g. of convergence or infinite sums within Algebra.
A professor whose math class I took said something like "To see at once the beauty of the calculus, observe how it bridges between algebra and geometry". I liked that. His PhD was in Physics, but he was a family man, with the associated pressing obligations, and the job he was offered by the university was teaching math. He said he thought he'd have preferred to teach physics, but he was happy teaching math, in large part because it had so much that was so beautiful.
 
  • #37
Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function and the plane that contains ...Integral Calculator
 

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