I've been doing some maths work using the rate of flow of liquids. I've used various models for the flow and various methods to integrate these models. The one thing that is confusing me is the difference between constants of proportionality, and constants of integration?


They are completely different animals. About the only thing they have in common is the words "constant of".

When two quantities, $p$ and $q$, are "proportional", that means that growth in one will be "mirrored" by growth in the other, and vice-versa. The growth need not be exactly the same (for example, your weight is [roughly] proportional to your height, but an increase in weight of 10 pounds does not correspond to an increase in height of 10 inches). There is a certain "scaling" that goes on. The "constant of proportionality" is what represents that scaling. We will have something of the form $p=kq$, with $k$ a constant; that means that for every unit that $q$ increases, $p$ will increase by $k$ units. That's the constant of proportionality. It is a constant, that depends on the quantities being considered.

When you do an indefinite integral, on the other hand, you are trying to find all antiderivatives of a function. That is, $\int f(x)dx$ represents a family of functions, namely, all functions $F(x)$ such that $F'(x)=f(x)$. Functions will generally have a lot of antiderivatives. But one can prove that if $F(x)$ and $G(x)$ are two functions such that $F'(x)=G'(x)=f(x)$, then $F$ and $G$ will just be vertical translates of each other; that is, there will exist a constant $c_0$ such that $G(x)=F(x)+c_0$. In order to represent all antiderivatives of a function $f(x)$, then, it is enough to find a single antiderivative $F(x)$, because then every other antiderivative will look like "$F(x)+c$" for some constant $c$. So we write $$\int f(x)dx = F(x)+C$$ to represent the entire family of antiderivatives; this means: "the antiderivatives of $f(x)$ are all the functions of the form $F(x)+C$, where $C$ is a constant." Pick a constant, you get an antiderivative. Pick a different constant, you get a different antiderivative. If $G$ is any antiderivative, then there will be some constant $C$ such that $G(x)=F(x)+C$. We call $C$ the "constant of integration." So $C$ is not a specific constant, but rather corresponds to the fact that antiderivatives are families, not specific functions.

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