I've recently started learning multivariable calculus. When reading about how to compute the length of a curve there is this formula:
$\int_{a}^{b} |\vec{r'}(t)| \,dt $
From my past understanding of integration, one should always introduce a arbitrary constant $C$, since the derivative of such a function that arbitrary constant is lost. Why is this not the case when integrating over the length of a curve?
An example from the litterature I'm reading:
"Find the length of the curve $\vec{r} = t^2\vec{i}+t^2\vec{j}+t^3\vec{k}$ from $t=0$ to $t=1$.
As a first step i'll take the derivative of $\vec{r}$.
$\vec{v}=\vec{r'}=2t\vec{i}+2t\vec{j}+3t^2\vec{k}$
Next I'll use the formula of finding the arc length.
$\int_{0}^{1} |\vec{r'}(t)| \,dt = $ $ \int_{0}^{1} \sqrt{8t^2+9t^4} \,dt = $ $ \int_{0}^{1} \sqrt{t^2(8+9t^2)} \,dt = $ $\int_{0}^{1} t\sqrt{8+9t^2} \,dt =\frac{1}{18}\frac{2}{3}*(8+9t^2)^{3/2}\bigg\rvert_{0}^{1}=1.7579$ (about this, more decimals of course).
This is correct with the answer in the book, but why do I not introduce an arbitrary constant when integrating? Is it simply because if we'd introduce such a constant, we'd not be able to figure out it's lenght? Given that we'd then have too many variables? Perhaps I have missed some key insight, or my mind is playing tricks..!
Let $f:[a,b] \to \mathbb R$ be a Riemann integrable function an $F$ an antiderivative of $f$. If $c$ is a constant, then $G:=F+c$ is also an antiderivative of $f$.
We get
$$\int_a^b f(x) dx= G(b)-G(a)= F(b)+c-(F(a)+c)=F(b)-F(a).$$
Do you see what happens wit the contant $c$ ?