This comes from the beginning chapter on line integrals in the book Mathematical Methods for Science Students:
Suppose $y=f(x)$ is a real single-valued monotonic continuous function of $x$ in some interval $x_1
. Then if $P(x,y)$ and $Q(x,y)$ are two real single-valued continuous functions of $x$ and $y$ for all points of C, the integrals $\int_C P(x,y)dx,\quad\int_C Q(x,y)dy$ and, more frequently, their sum $\int_C \Bigg\{P(x,y)dx+Q(x,y)dy\Bigg\}$ are called curvilinear integrals or line integrals, the path of integration C being along the curve $y = f(x)$ from A to B.
Is this a correct definition of a line integral?
It doesn't appear to resemble the wikipedia definition of a line integral over a scalar field, which makes sense to me:
For some scalar field $f : U\subseteq R^n → R$, the line integral along a piecewise smooth curve $C \subset U$ is defined as$\int_C f ds = \int^b_af(r(t))|r'(t)|dt$ where r: [a, b] → C is an arbitrary bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C and a
The function f is called the integrand, the curve C is the domain of integration, and the symbol ds may be intuitively interpreted as an elementary arc length. Line integrals of scalar fields over a curve C do not depend on the chosen parametrization r of C.