Line Integrals Introduction

Parametric Curves Review

A curve in the plane (or space) is described by a vector function $\mathbf{r}(t) = (x(t),\, y(t))$ as $t$ varies over an interval. The velocity vector $\mathbf{r}'(t)$ is tangent to the curve at each point.

Parameter $t$:

$t = 1.57$ | $\mathbf{r}(t) = (0.00, 1.00)$ | $\mathbf{r}'(t) = (-1.00, 0.00)$

Speed: $\|\mathbf{r}'(t)\| = \sqrt{x'(t)^2 + y'(t)^2}$ - this measures how fast the point moves along the curve.

Beyond Parametric Curves

With parametrization in hand, the three flavors of line integral - $\int_C f\,ds$ (arc length form), $\int_C P\,dx + Q\,dy$ (differential form), $\int_C \mathbf{u}\cdot d\mathbf{r}$ (vector form) - all reduce to ordinary one-variable integrals over the parameter $t$. Each gets a dedicated chapter page that builds the formula from first principles and shows worked examples:

§5.1 Introduction to Vector Integral Calculus - the three types side by side
§5.2 Line Integrals in the Plane - $\int P\,dx + Q\,dy$, direction, closed curves
§5.3 Line Integrals w.r.t. Arc Length - the curtain interpretation
§5.4 Line Integrals as Integrals of Vectors - work, circulation, conservative fields

That's where the depth lives. The card above (parametrization) is the only piece you genuinely need before opening any of those four pages.