Scalar Line Integrals: Let x : [a, b] → R^{3} be a C^{1} path. Δs_{k} = length of kth segment of path

= ∫_{t_{k−1}}^{t_{k}} ||x′(t)||dt = ||x′(t_{k}^{∗∗})||(t_{k} − t_{k−1}) = ||x′(t_{k}^{∗∗})||Δt_{k}

for some t_{k}^{∗∗} ∈ [t_{k−1}, t_{k}]

∫_{x} f ds ∼ Σ_{i=1}^{n}f(x(t_{k}^{∗}))Δs_{k} = Σ_{i=1}^{n}f(x(t_{k}^{∗}))||x′(t_{k}^{∗∗})||Δt_{k}

= ∫_{a}^{b} f(x(t))||x′(t)||dt

Vector Line integrals: Calc 1 review: Suppose f : R → R, ∫_{a}^{b} f′(t)dt = f(b) − f(a). ∫_{a}^{b}(velocity) dt = distance traveled. ∫_{a}^{b}(rate of change)dt = total change. Given f′(t), can find f(b) − f(a). Given velocity, can find distance traveled. Given rate of change, can find total change. Calc III: Suppose f : R^{2} → R, Given ∇f = (\frac{∂f}{∂x}, \frac{∂f}{∂y}), find f(p) − f(q).