Can someone explain to me why the integrand of line integral is $f(x,y,z)=1$ when trying to find the length of the curve? Click the link below to access the image/ example for reference purposes
Flight of an eagle An eagle soars on the ascending spiral path $$C\colon \mathbf r(t) = \langle x(t),y(t),z(t)\rangle = \left\langle 2400cos\frac t2, 2400\sin\frac t2, 500t\right\rangle,$$ where $x$, $y$ and $z$ are measured in feet and $t$ is measured in minutes. How far does the eagle travel over $0\le t\le 10$?
Because we are finding the length of a curve, the integrand in this integral is $f(x,y,z)=1$.
This is for essentially the same reason that for $a < b$ we can find the length of interval $[a,b]$ via $$\int_{a}^{b} \,dx = b - a.$$ Let's consider the case of a curve $C$. If we want to compute it's length $L$, one way we can do that is to think of approximating the length of the curve by breaking the curve into many small line segments. We can first approximate the length in this manner by sampling $n+1$ points $(x_{i},y_{i},z_{i})$ along the curve, and finding the distance between two consecutive points $(x_{i},y_{i},z_{i})$ and $(x_{i+1},y_{i+1},z_{i+1})$ via $$\Delta s_{i} = \sqrt{(x_{i+1} - x_{i})^{2} + (y_{i+1} - y_{i})^{2} + (y_{i+1} - y_{i})^{2}}.$$ Then, we can say that $$L \approx \sum_{i = 1}^{n}\Delta s_{i}.$$ If we take the limit as the size of the greatest distance between consecutive points goes to zero we get the actual length: $$L = \lim_{\|\Delta s\| \to 0}\sum_{i = 1}^{n}\Delta s_{i}.$$ Note, however, that we can rewrite this as $$L = \lim_{\|\Delta s\| \to 0}\sum_{i = 1}^{n}\Delta s_{i} = \lim_{\|\Delta s\| \to 0}\sum_{i = 1}^{n}1\cdot \Delta s_{i} = \int_{C}\,ds.$$