Line Integrals

The central idea

For a piecewise $C^1$ curve $\mathbf r:[a,b]\to\mathbb R^n$, the scalar line integral is $\int_C fds=\int_a^bf(\mathbf r(t))\|\mathbf r'(t)\|dt$, independent of orientation. For a vector field, work is $\int_C\mathbf F\cdot d\mathbf r=\int_a^b\mathbf F(\mathbf r(t))\cdot\mathbf r'(t)dt$, which changes sign when orientation reverses.

Definitions, hypotheses, and notation

A parametrization may vary speed without changing either integral, provided it traces the curve once with the required orientation. In the scalar case, the speed norm absorbs reparametrization and removes orientation. In the vector case, the signed derivative preserves orientation. Repeated tracing multiplies the accumulated result.

For $\mathbf F=\langle P,Q,R\rangle$ in three dimensions, work may also be written $\int_C Pdx+Qdy+Rdz$. This differential notation is shorthand for the same parametrized dot product, not permission to integrate components independently without the curve relation.

Conceptual meaning

A line integral accumulates along a one-dimensional curve embedded in higher-dimensional space. The scalar version weights arc length; the vector version projects the field onto the oriented tangent displacement and measures work or circulation.

A dependable method and decision rule

1. Choose a parametrization with stated bounds and orientation.
2. Compute $\mathbf r'$ and either its norm or its dot product with the field.
3. Substitute the curve into every field or density component.
4. Integrate over the parameter interval.
5. Check orientation dependence and whether a potential offers a shortcut.

Fully worked example

Graphical or geometric meaning

At each point, only the field component parallel to the curve's oriented tangent contributes to work. Perpendicular field arrows give zero dot product even if their magnitudes are large.

Common mistakes and why they fail

Verification and reasonableness checks

• Verify parameter bounds produce correct endpoints.
• Reverse the curve and confirm vector work negates.
• Use a potential when available as an independent evaluation.

Identify what is being accumulated

A scalar line integral $\int_C f\,ds$ accumulates density along length and is independent of orientation; use $ds=\|\mathbf r'(t)\|dt$. A vector work integral $\int_C\mathbf F\cdot d\mathbf r$ uses $\mathbf F(\mathbf r(t))\cdot\mathbf r'(t)dt$ and changes sign when orientation reverses. Confusing these differentials is a common source of missing speed factors. Parameterize the intended path once, include its direction, and convert every coordinate in the integrand. For work, a conservative field may reduce the calculation to endpoints; otherwise evaluate directly or use an applicable theorem. Check units and sign against the physical interpretation of density, circulation, or work.

Practice

1. Find $\int_C1ds$ for a line segment of length $5$.
2. Find work of constant $\mathbf F=\langle2,0\rangle$ from $(0,0)$ to $(3,4)$.
3. What happens to work under reversed orientation?

Connections and next steps

• Fundamental Theorem for Line Integrals
• Vector Fields
• Green's Theorem