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?
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$\begingroup$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 $a$.
$\endgroup$ 1 $\begingroup$The traditional notion is that to evaluate $\int_a^b f(x) dx$, the x axis is the horizontal axis, the $f(x)$ axis or $y$ axis [if you presume that $y = f(x)$] is the vertical axis, and the integral represents the area under the curve in the region bounded by $x=a$, and $x=b$.
When you make a $u$ substitution, this simply means that you have transformed the original integral into something that looks like $\int_c^d g(u) du$, where the u axis is the horizontal axis, the $g(u)$ axis is the vertical axis, and the integral represents the area under the curve in the region bounded by $u = c$ and $u = d.$
By the way, one can construe all of the above as vertical integration, re you are looking for the area under the curve, rather than the area to the left of the curve.
Suppose that you are given $y = f(x)$, and you are asked for the area to the left of the curve in the relevant region. You then have two choices. Compute $g(y) = x$ (if feasible) where $g$ is the inverse function of $f$. Then it would make sense to set up an integral that looks like $\int_e^f g(y) dy$, where $e,f$ represent the corresponding $y$-value bounds for the integral. I would construe this to be integrating horizontally, because you are looking for the area to the left of the curve $y = f(x).$
The alternative and often much easier approach is to attack the problem simply by insisting on computing the area under the curve (i.e. vertical integration). Then, assuming that you have precisely identified the length and width of the pertinent rectangle, and have determined that the area of this rectangle $= R$, then you have that the (area to the left of the curve) + the (area under the curve) = R.
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