Interpreting integration otherwise

[archaic]

There's one odd way to think about integration when it comes to interpreting it as a sum. suppose for a second that $x$ is in meters, you could think of distance as an infinite number $n$ of "points" in space, $n→∞$, then in this case $f(x)Δx$ would mean that you now have $nf(x)$.

Now, as usual, you think of $Δx$ as vanishing to $0$, and then think of $f(x)dx[meters]$ as just, nearly, one point times $f(x)$, and so summing all these would get you the sum of all $f(x)$s, in meters that is, so if the integral equals $5.36$, you consider it as the number of points in $5.36$ meters, which is infinite of course, but it's a way of "summing" infinity.

If you think of it this way, it kind of also seems directly deducible that the average of $f(x)$ is $\int_a^b\frac{1}{b-a}f(x)dx$, you're diving by the "number of points" in $b−a$ meters, which corresponds to the number of $f(x)$s. Again, infinite of course, but we're just manipulating infinities using names.

How bizarre is this, and what remarks could you make for such a way of thinking about integrals? Where can this pose problems?

 

[fresh_42]

You cannot treat dimensions like that. A point has no volume hence you cannot "add" infinitely many to get a length. The dimensions and our geometric intuition don't match. You basically said: $0\cdot \infty = 5.36\,m$ which is nonsense. That is why we use limits and the $\varepsilon-$notation instead. Neither is zero an element in the multiplicative group of $\mathbb{R}$ nor is infinity a real number at all. So if you neglect both facts, you get of course weird results: Every statement can be deduced from a false one.

 

[jedishrfu]

Visualizing integration as a sum of thin rectangles papering an area is a well known way of thinking about it. However, you can’t just wave your hands and reduce dx to a “single point” as that has no meaning mathematically that’s why we think in terms of limits and series to integrate.

The YouTube channel, 3blue1brown has several good videos that help to visualize calculus concepts and I would suggest you watch them.

 

[Mark44]

that’s why we honk in terms of limits and series to integrate.

honk?

Did you mean think? If not, I have no idea what you meant.

 

[Mark44]

There's one odd way to think about integration when it comes to interpreting it as a sum. suppose for a second that $x$ is in meters, you could think of distance as an infinite number $n$ of "points" in space, $n→∞$, then in this case $f(x)Δx$ would mean that you now have $nf(x)$.

As @fresh_42 already said, this is the wrong way to think about things, but I think it bears repeating -- points have zero length, so $n f(x)$ is meaningless.

 

[jedishrfu]

honk?

Did you mean think? If not, I have no idea what you meant.

I’m having trouble with spell checking. It appears to be overaggressive in correctly things as I type and I’m not sure how to fix it yet. Part of the problem may be related to the new site software and iOS although I don’t know how.

 

[Mark44]

It appears to be overaggressive in correctly things as I type and I’m not sure how to fix it yet.

Is there an option to turn it off? Or turn off autocompletion?

 

[FactChecker]

If you want to see another approach to integration that has real advantages, you might look at Lebesgue integration and measure theory. It is like Riemann integration turned on its side.

 

[fresh_42]

If you want to see another approach to integration that has real advantages, you might look at Lebesgue integration and measure theory. It is like Riemann integration turned on its side.

Good idea! See https://www.physicsforums.com/insights/omissions-mathematics-education-gauge-integration/