I Geometric meaning of reversing the limits of an integral

[sysprog]

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".

 

[robphy]

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.

 

[WWGD]

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.

 

[WWGD]

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.

 

[sysprog]

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.

 

[Stephen Tashi]

The absolute value comes in with the word volume.

@mech-eng : Here's a related thread:
https://www.physicsforums.com/threads/change-of-variable-and-jacobians.905474/#post-5702972

 

[DennisHailey]

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

 

[fresh_42]

...Integral Calculator

See also
https://www.physicsforums.com/threa...h-physics-earth-and-other-curiosities.970262/