∞Calculus IV Unit 18 – Line Integrals

Key Concepts

• Line integrals evaluate the integral of a function along a curve or path in a plane or space
• Fundamental objects include a vector field $\mathbf{F}(x, y, z)$ and a parameterized curve $\mathbf{r}(t) = (x(t), y(t), z(t))$
• Line integrals have two main types: scalar line integrals and vector line integrals
  • Scalar line integrals integrate a scalar function along a curve
  • Vector line integrals integrate a vector field along a curve
• The Fundamental Theorem of Line Integrals relates the line integral of a gradient field to the values of a potential function at the endpoints of the curve
• Green's Theorem relates a line integral around a simple closed curve to a double integral over the region bounded by the curve
• Line integrals have applications in physics (work, circulation) and engineering (flux, flow)

Definition and Intuition

• A line integral is an integral where the function to be integrated is evaluated along a curve
• Intuition: line integrals measure the accumulation of a quantity (scalar or vector) along a path
• Denoted as $\int_C f(x, y, z) \, ds$ for scalar line integrals or $\int_C \mathbf{F} \cdot d\mathbf{r}$ for vector line integrals
  • $C$ represents the curve
  • $ds$ represents an infinitesimal arc length along the curve
  • $d\mathbf{r}$ represents an infinitesimal displacement vector along the curve
• Geometrically, a scalar line integral can be interpreted as the area under the curve $f(x, y, z)$ along the path $C$
• A vector line integral measures the work done by a force field $\mathbf{F}$ along the curve $C$

Types of Line Integrals

• Scalar line integrals: $\int_C f(x, y, z) \, ds$
  • Integrate a scalar function $f(x, y, z)$ along a curve $C$
  • Example: $\int_C (x^2 + y^2) \, ds$ (integrating the function $f(x, y) = x^2 + y^2$ along $C$)
• Vector line integrals: $\int_C \mathbf{F} \cdot d\mathbf{r}$
  • Integrate a vector field $\mathbf{F}(x, y, z)$ along a curve $C$
  • Two types: line integrals with respect to arc length and line integrals with respect to coordinates
    • Line integrals with respect to arc length: $\int_C \mathbf{F} \cdot \mathbf{T} \, ds$, where $\mathbf{T}$ is the unit tangent vector to the curve
    • Line integrals with respect to coordinates: $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_C P \, dx + Q \, dy + R \, dz$, where $\mathbf{F} = (P, Q, R)$
• Line integrals can be evaluated over closed curves (line integrals around a closed path) or open curves (line integrals from one point to another)

Calculating Line Integrals

• Parameterize the curve $C$ using a vector function $\mathbf{r}(t) = (x(t), y(t), z(t))$, where $a \leq t \leq b$
• For scalar line integrals:
  • Substitute the parameterization into the function $f(x, y, z)$
  • Calculate the arc length differential: $ds = |\mathbf{r}'(t)| \, dt = \sqrt{(dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2} \, dt$
  • Integrate with respect to the parameter $t$: $\int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)| \, dt$
• For vector line integrals with respect to arc length:
  • Calculate the unit tangent vector: $\mathbf{T}(t) = \mathbf{r}'(t) / |\mathbf{r}'(t)|$
  • Dot the vector field with the unit tangent vector: $\mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{T}(t)$
  • Integrate with respect to the parameter $t$: $\int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{T}(t) |\mathbf{r}'(t)| \, dt$
• For vector line integrals with respect to coordinates:
  • Substitute the parameterization into the vector field components: $P(x(t), y(t), z(t))$, $Q(x(t), y(t), z(t))$, $R(x(t), y(t), z(t))$
  • Calculate the coordinate differentials: $dx = (dx/dt) \, dt$, $dy = (dy/dt) \, dt$, $dz = (dz/dt) \, dt$
  • Integrate with respect to the parameter $t$: $\int_a^b P(x(t), y(t), z(t)) \frac{dx}{dt} + Q(x(t), y(t), z(t)) \frac{dy}{dt} + R(x(t), y(t), z(t)) \frac{dz}{dt} \, dt$

Properties and Theorems

• Linearity: For scalar functions $f$ and $g$ and constants $a$ and $b$, $\int_C (af + bg) \, ds = a \int_C f \, ds + b \int_C g \, ds$
• Additivity: If a curve $C$ is split into two parts $C_1$ and $C_2$, then $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_{C_1} \mathbf{F} \cdot d\mathbf{r} + \int_{C_2} \mathbf{F} \cdot d\mathbf{r}$
• Fundamental Theorem of Line Integrals: If $\mathbf{F} = \nabla f$ is a conservative vector field, then $\int_C \mathbf{F} \cdot d\mathbf{r} = f(\mathbf{r}(b)) - f(\mathbf{r}(a))$, where $\mathbf{r}(a)$ and $\mathbf{r}(b)$ are the endpoints of the curve $C$
  • Consequence: The line integral of a conservative vector field is path-independent
• Green's Theorem: Relates a line integral around a simple closed curve $C$ to a double integral over the region $D$ bounded by $C$
  • $\oint_C P \, dx + Q \, dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \, dA$
  • Useful for converting line integrals to double integrals and vice versa
• Stokes' Theorem: Generalizes Green's Theorem to higher dimensions, relating the surface integral of the curl of a vector field over a surface to the line integral of the vector field around the boundary of the surface

Applications in Physics and Engineering

• Work: The line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ represents the work done by a force field $\mathbf{F}$ along a curve $C$
  • Example: Work done by gravity on an object moving along a path
• Circulation: The line integral $\oint_C \mathbf{F} \cdot d\mathbf{r}$ represents the circulation of a vector field $\mathbf{F}$ around a closed curve $C$
  • Example: Circulation of a fluid velocity field around a closed path
• Flux: The surface integral $\iint_S \mathbf{F} \cdot d\mathbf{S}$ represents the flux of a vector field $\mathbf{F}$ through a surface $S$
  • Related to line integrals via Stokes' Theorem
  • Example: Electric flux through a surface in an electric field
• Flow: Line integrals can be used to calculate the flow of a fluid along a curve
  • Example: Mass flow rate of a fluid through a pipe
• Potential difference: The line integral of an electric field along a path gives the potential difference (voltage) between the endpoints
  • $V_b - V_a = -\int_a^b \mathbf{E} \cdot d\mathbf{r}$

Common Pitfalls and Tips

• Choosing the wrong parameterization can make the integral more difficult to evaluate
  • Try to choose a parameterization that simplifies the integrand
• Forgetting to calculate the arc length differential ($ds$) or coordinate differentials ($dx$, $dy$, $dz$)
  • Always include these terms in the integral
• Misinterpreting the meaning of the line integral
  • Be clear about what the integral represents in the context of the problem (work, circulation, flux, etc.)
• Incorrectly applying the Fundamental Theorem of Line Integrals
  • The theorem only applies to conservative vector fields (fields that are the gradient of a scalar potential function)
• Not checking the continuity and differentiability of the functions involved
  • Line integrals require the functions to be continuous and differentiable along the curve
• Tips:
  • Sketch the curve and the vector field (if applicable) to visualize the problem
  • Break the curve into smaller, simpler segments if needed
  • Use symmetry or other properties of the curve and vector field to simplify the integral
  • Check your answer using alternative methods (e.g., Green's Theorem) when possible

Practice Problems and Examples

1. Evaluate the line integral $\int_C (x + y) \, ds$, where $C$ is the straight line segment from $(0, 0)$ to $(1, 1)$.
   • Parameterize the line: $\mathbf{r}(t) = (t, t)$, $0 \leq t \leq 1$
   • $ds = \sqrt{2} \, dt$
   • $\int_C (x + y) \, ds = \int_0^1 (t + t) \sqrt{2} \, dt = \sqrt{2}$
2. Calculate the work done by the force field $\mathbf{F}(x, y) = (x^2, y)$ along the curve $C$ given by $y = x^2$ from $(0, 0)$ to $(1, 1)$.
   • Parameterize the curve: $\mathbf{r}(t) = (t, t^2)$, $0 \leq t \leq 1$
   • $\mathbf{F}(\mathbf{r}(t)) = (t^2, t^2)$
   • $d\mathbf{r} = (1, 2t) \, dt$
   • $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_0^1 (t^2 \cdot 1 + t^2 \cdot 2t) \, dt = \frac{7}{6}$
3. Use Green's Theorem to evaluate the line integral $\oint_C (2x + y^2) \, dx + (x^2 - 2y) \, dy$, where $C$ is the boundary of the region enclosed by the parabola $y = x^2$ and the line $y = 4$.
   • Identify $P(x, y) = 2x + y^2$ and $Q(x, y) = x^2 - 2y$
   • $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 2x - 2y$
   • Use Green's Theorem: $\oint_C (2x + y^2) \, dx + (x^2 - 2y) \, dy = \iint_D (2x - 2y) \, dA$
   • Evaluate the double integral over the region $D$ bounded by $y = x^2$ and $y = 4$
4. Determine whether the vector field $\mathbf{F}(x, y) = (y \cos x, \sin x)$ is conservative, and if so, find a potential function $f(x, y)$ such that $\mathbf{F} = \nabla f$.
   • Check if $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$
   • $\frac{\partial P}{\partial y} = \cos x$ and $\frac{\partial Q}{\partial x} = \cos x$, so $\mathbf{F}$ is conservative
   • To find a potential function, integrate $P$ with respect to $x$: $f(x, y) = \int y \cos x \, dx = y \sin x + C(y)$
   • Differentiate $f$ with respect to $y$ and compare with $Q$: $\frac{\partial f}{\partial y} = \sin x + C'(y) = \sin x$, so $C(y) = 0$
   • A potential function is $f(x, y) = y \sin x$
5. Verify Stokes' Theorem for the vector field $\mathbf{F}(x, y, z) = (y^2, xz, x^2)$ and the surface $S$ given by $z = 4 - x^2 - y^2$, $z \geq 0$, oriented upward.
   • Stokes' Theorem states that $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}$, where $C$ is the boundary of $S$
   • Calculate $\nabla \times \mathbf{F} = (2x, -2y, z - x)$
   • Parameterize the surface: $\mathbf{r}(r, \theta) = (r \cos \theta, r \sin \theta, 4 - r^2)$, $0 \leq r \leq 2$, $0 \leq \theta \leq 2\pi$
   • $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \int_0^{2\pi} \int_0^2 (2r \cos \theta, -2r \sin \theta, 4 - r^2 - r \cos \theta) \cdot (r \cos \theta, r \sin \theta, -2r) \, dr \, d\theta = -8\pi$
   • Parameterize the boundary curve: $\mathbf{r}(\theta) = (2 \cos \theta, 2 \sin \theta, 0)$, $0 \leq \theta \leq 2\pi$
   • $\oint_C \mathbf{F}