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Line integrals ∫_C f ds (scalar) or ∫_C F · dr (vector) integrate along curves. The scalar version sums f weighted by arc length; the vector version computes work done by F along a path C. Both depend on the parametrization's orientation.
Ordinary integrals accumulate a quantity along a straight line segment (the x-axis). Line integrals do the same thing along an arbitrary curve in space. The curve is the domain of integration, and you need a way to measure "how much" of that curve passes through each point. That is the role of arc length from your prerequisites: the scalar ds is an infinitesimal piece of arc length, telling you how long a tiny piece of the curve is.
The scalar line integral ∫_C f ds answers: if f(x,y,z) is a density or weight at each point, what is the total accumulated quantity along the curve? Imagine a wire whose linear density (mass per unit length) varies from point to point. The total mass is ∫_C ρ ds — sum up density times length element at each point along the wire. To compute this, you parametrize the curve: let r(t) for t ∈ [a,b] trace out C, then ds = |r′(t)| dt, and the integral becomes ∫_ab f(r(t)) |r′(t)| dt — a standard single-variable integral. The factor |r′(t)| is the speed of the parametrization, which converts the parameter increment dt into actual arc length.
The vector line integral ∫_C F · dr asks a different question: how much does the vector field F push the path forward? The integrand F · dr picks up the component of F in the direction of motion along C. This is the work done by a force field F on a particle traveling along C. Using the parametrization, dr = r′(t) dt, so the integral becomes ∫_ab F(r(t)) · r′(t) dt. The dot product F · r′(t) extracts how much F aligns with the direction of travel at each moment; integrating it gives cumulative work. When F is perpendicular to the path everywhere, this integral is zero — a force perpendicular to motion does no work.
Orientation matters for vector line integrals but not for scalar ones. Reversing the direction of traversal negates the vector line integral (since r′(t) reverses), but leaves the scalar integral unchanged (since |r′(t)| is always positive). This asymmetry reflects the underlying physics: a force field that aids your journey one way opposes it the other way. For conservative vector fields — those with a potential function — the vector line integral depends only on the endpoints, not the path taken. That is the content of the Fundamental Theorem for Line Integrals, which you will explore next.