6.2 Line Integrals

Learning Objectives

• 6.2.1: Calculate a scalar line integral along a curve.
• 6.2.2: Calculate a vector line integral along an oriented curve in space.
• 6.2.3: Use a line integral to compute the work done in moving an object along a curve in a vector field.
• 6.2.4: Describe the flux and circulation of a vector field.

Introduction to Line Integrals

• Line integrals extend the concept of integration to curves in the plane or space.
• They are applicable in engineering and physics, providing generalizations of the Fundamental Theorem of Calculus.
• Connected to vector fields, used for work done by a force field.

Scalar Line Integrals

• Integrals of a scalar function over a curve.
• Definition: Integrate a function over a curve, partition curve into small pieces.
• Use a Riemann sum approach, multiply function value by arc length of the piece, take a limit.
• Scalar line integral notation: ( \int_C f(x,y,z) ds ).

Formal Definition

• Let ( C ) be a smooth curve parameterized by ( r(t) = (x(t), y(t), z(t)) ).
• Scalar line integral: ( \int_C f(x,y,z) ds = \lim_{n \to \infty} \sum_{i=1}^n f(P_i^*) s_i ).
• For a planar curve: similar definition with two variable function ( f(x,y) ).*

Calculation

• Convert line integral into a parameterized integral: ( \int_a^b f(r(t)) |r'(t)| dt ).
• Arc length calculations can be done through line integrals where ( f(x,y,z) = 1 ).

Vector Line Integrals

• Integrate along a curve through a vector field.
• Requires orientation of curve.
• Used to find work done by a force field along a path.

Definition

• Work done by a force field ( \mathbf{F} ): ( W = \int_C \mathbf{F} \cdot \mathbf{T} , ds ).
• Vector line integral notation: ( \int_C \mathbf{F} \cdot d\mathbf{r} ).

Evaluation

• Translate to parameterization: ( \int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(r(t)) \cdot r'(t) , dt ).
• Reversing curve orientation changes the sign of the integral.

Flux and Circulation

• Flux: Measures flow across a curve, calculated as ( \int_C \mathbf{F} \cdot , \mathbf{N} , ds ).
• Circulation: Line integral of ( \mathbf{F} ) along a closed curve (denoted ( \oint_C \mathbf{F} \cdot \mathbf{T} , ds )).
• Used in characterizing fluid flow and conservative fields.

Applications

• Scalar Line Integrals: Calculate mass of a wire, surface area of a sheet.
• Vector Line Integrals: Calculate work done by a force field or fluid flow rate across a curve.

Examples

• Mass Calculation: Using density function to find total mass along a parameterized curve.
• Work Calculation: Compute work done by a vector field along a given path.

Section Exercises

• Practice problems involve computing line integrals for various parameterizations and conditions.
• Exercises also cover work computations in different vector fields and flux calculations across curves.