When dealing with derivatives, "with respect to $x$" means we are observing how a small change in $x$ (the input) affects a change in $y$ (the output).
I found this conceptualization very helpful and it made other derivative related concepts feel more intuitive.
I'm wondering if there is a similar conceptualization of what "with respect to $x$" means when integrating. In particular, how does the input, $x$, affect or relate to the output, $y$, when integrating?
I should say that I'm familiar with the geometric conceptualization of an integral, namely the Riemann sum, and that integrating with respect to $x$ means using the $x$-axis as the lower bound (or base) of the curve when calculating area. Alternatively, one can integrate with respect to $y$ and then the $y$-axis is used as a bound instead. However, it is difficult for me to glean from the geometric interpretation what "with respect to $x$" means when integrating.
This question is motivated by using $u$-substitution requires integration with respect to $u$, but there is no $u$ axis to use as a base to find the area with. I'm sure my understanding of this is incorrect, hence why I'm hoping that better understanding what "with respect to __" means when integrating will help me better understand u-substitution and other integration concepts, much like how understanding what "with respect to __" means when differentiating helped me better understand the Chain Rule.
In shot my main question is:
What does "with respect to __" mean when integrating, as in how does the input affect or relate to the output when finding the area under the curve? Is there a conceptualization along similar lines to what "with respect to __" means when differentiating?
Best Answer
A lot of integral formulas have other variables than $x$ floating around inside them, e.g. $$\int \frac{k dx}{x^2 + a^2} = \frac{k}{a} \arctan \frac{x}{a} + C,$$ and the $dx$ formalism is necessary to specify which of the variables is the dummy variable of integration. We say that the above integral is taken "with respect to $x$" to clarify that it is not being taken with respect to $k$ or to $a$.