I Integration with different infinitesimal intervals

[JohnnyGui]

It is questionable to use the integration variable $t$ indicated by $dt$ under the integral anywhere else.

The integral reads $\int_a^b f(t)\,dt =\int_{t=a}^{t=b} f(t)\,dt.$ This resulted in the equation $t=x+\Delta t$ for the upper bound as used in your linked article. It is disturbing to have the same variable $t$ in one equation but with two meanings! $t=x+\Delta x$ would have been the better choice.

It is further problematic to substitute $\Delta x$ by $\delta x$ or $dx.$ They have different meanings, even in case we consider $\Delta x \to 0.$ The understanding of $dx$ in their various contexts is difficult enough even without adding another context.

Thank you.

How about the upper limit of the equation in pbuk's post of this thread if we substitute $\delta x$ by $\Delta x$ giving:
$$\lim_{\Delta x \to 0} \int_a^{a+\Delta x} f(x) dx \approx \lim_{\Delta x \to 0} f(a) \Delta x$$
I'd expect you won't agree with this upper limit, even if $a$ is considered a value and not a variable? After all, $x=a+\Delta x$ would also show different meanings for $x$.

 

[fresh_42]

Thank you.

How about the upper limit of the equation in pbuk's post of this thread if we substitute $\delta x$ by $\Delta x$ giving:
$$\lim_{\Delta x \to 0} \int_a^{a+\Delta x} f(x) dx \approx \lim_{\Delta x \to 0} f(a) \Delta x$$
I'd expect you won't agree with this upper limit, even if $a$ is considered a value and not a variable? After all, $x=a+\Delta x$ would also show different meanings for $x$.

It breaches my first rule: never use the integration variable elsewhere. I would write it as
$$\lim_{\Delta a \to 0} \int_a^{a+\Delta a} f(x) dx \approx \lim_{\Delta a \to 0} f(a) \Delta a$$
Why use $x$ if we already have $a$ for values on the $x$-axis? If you don't like the same $a$ in $a+\Delta a$ then $\Delta a = h$ is your letter of choice.

However, this rule is for clarity. As mentioned earlier $\int_d^{d+\Delta d}f(d)dd$ is possible to write. We use the same letter for quite a couple of different meanings and its position provides context and meaning. You would certainly agree that such a formula is a nightmare rather than an integral. You could do it, but I strongly recommend not. Same with the $x$ in its double role as an integration (dummy) variable and as an upper limit. You can do it, but it causes threads like this one with meanwhile 30+ posts.

More important is the idea behind the formula. Assume that $F$ is an anti-derivative of $f,$ i.e. $F'=f.$ Let further be $F(x)=F(a)+F'(a)(x-a)+ O((x-a)^2)$ the Taylor series of $F.$ Then
\begin{align*}
\lim_{\Delta a \to 0} \int_a^{a+\Delta a} f(x) dx&=\lim_{\Delta a \to 0}( F(a+\Delta a)-F(a))\\
&=\lim_{\Delta a \to 0}(F(a)+F'(a)\cdot(a+\Delta a -a)+ O((a+\Delta a -a)^2) - F(a))\\
&=\lim_{\Delta a \to 0}(f(a)\cdot \Delta a +O((\Delta a)^2))\\
&\approx\lim_{\Delta a \to 0}f(a)\cdot \Delta a
\end{align*}
Imagine if I had used $\Delta x$ instead of $\Delta a.$ How would you know the difference to the $x$ I used as a variable in the Taylor series? It would open up another discussion about naming objects in mathematics. I like the motto: different meaning requires different letter.