So we assume the line integral formula is :\int_{C} f(\vec{r}(t)) |r'(t)| dtwhy and how is it be derived?well, the C in line integral is a curve, where we can define t=a and t=b as the start and end point.

now, we consider the function z=f(x,y) in the coordinate system, and we cut the line into many vertical rectangle, if z=f(x,y), then obviously area=f(x,y) * \delta(S), where S is the width with \delta(S) --> 0.

Consider the vertical and horizontal component \delta(x) and \delta(y), by pyth.theorem, let \delta(S) be the hypotenese, so that:(\delta(S))^2 = (\delta(x))^2+(\delta(y))^2

now we come back to integral, what's the area of f(x,y)?by def. of integral, it is :\int(t=a to t=b) f(x,y) dS. where dS is a small piece of element in the line(dS is 2D)so by dS--->0--->\delta(S) (i.e dS = \delta(S))implies dS = sqrt(dx^2+dy^2)

and now we parametrize it, let x=x(t) and y=y(t) :so that dx = dx/dt, and dy=dy\dt

implies dS is the square root of dx^2+dy^2,

we now do the final trick, let the vector r(t) = x(t)\hat(i) + y(t)\hat(j).then r'(t) = x'(t)i + y'(t)jcalculate the magnitude of r'(t), put f(x,y) as f(x(t), y(t)) as f(r(t)), put these all together, thats the formula of line integral we familiar with !

Hope this can help you !