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Suppose I define $G(x)=\int_0^xf(t)dt$ as the area under the function $f(t)$ of which both $0$ and $x$ are within the domain of $f$, and $x$ can take any value within the domain.

Hence we can conclude $x$ is not a constant since it can take any values and also the area under the graph changes for different values of x.

Next, define $\int_0^bg(t)dt$ as the area under $g(t)$ such that $0$ and $b$ are within the domain of $g$, and $b$ is any constant value. (i.e it can be a $5$, a $13$, $\pi$, etc.)

I know that if $b$ is $5$, $13$, $\pi$ then $\int_0^5g(t)dt$, $\int_0^{13}g(t)dt$, $\int_0^{\pi}g(t)dt$ or the areas under these specific constants are constants. However my intuition tells me that since $b$ can take any constant values, $b$ itself isn't constant. Is this argument valid? Maybe it's because of my definition that $b$ can take any constant values that made it "not a constant". I know it is more than usual even in textbooks to assume $b$ in $\int_0^bf(t)dt$ is constant. And that, the author or the writer must have meant that "I'll give you a specific value, but I won't tell you what it is."

Because of this question I'm afraid I'm on the wrong path now towards learning mathematics. I'm a senior HS student (not living in US) planning to take an engineering degree, but my curiosity about the mathematics behind it especially calculus is like that of any aspiring mathematician.

Bahog Bilat
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    "variable", "parameter", "constant" are tems fo rthe same thing, the different levels of constantness are only in our minds. For example, the value of $x$ in $G(x)=\int_0^xf(t),dt$ does not change while we move our eyes fromth eleft to the right side of the equation. The value of a (constant? parameter? variable?) $b$ may be considered the same during a multi-page argument. The value of $\pi$ never changes (until you no longer mean the circumference-diameter ration, but instead use the same symbol to denote an arbitrary permutation or the prime counting function) – Hagen von Eitzen Dec 27 '17 at 07:23

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Your question highlights a confusion that may arise when we do not distinguish between a variable or changing quantity, and an undetermined or unspecified constant quantity.

If a non-numerical symbol, say $ b, $ is called a constant, then it is to be understood as non-varying, only that the particular constant is not concretely or numerically specified or defined.

Essentially then, you're wondering whether an indeterminate constant isn't a variable. Well, strictly speaking they're not the same since one is defined as changing while the other is stable, but most people -- wrongly -- don't differentiate between them.

This is especially transparent when dealing with formal manipulations; it is necessary to differentiate between an indeterminate constant and a variable. This will guide our treatment towards the symbols and eliminate confusions. This is a context where differentiating between the two is not only useful, but logically necessary. Consider, for example, the polynomial in the indeterminate $ y $ -- now many would say in the variable $ y $ instead. While this makes no great visual difference, there is a great conceptual difference, and this is another reason why some confuse polynomials and functions defined by polynomials. They are not the same at all.

Allawonder
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