Chapter 16 – Vector Calculus 16.2 Line Integrals Objectives: Understand various aspects of line integrals in planes, space, and vector fields Dr. Erickson 16.2 Line Integrals

Line Integrals They were invented in the early 19th century to solve problems involving: Fluid flow Forces Electricity Magnetism Dr. Erickson 16.2 Line Integrals

Line Integrals We start with a plane curve C given by the parametric equations (Equation 1) x = x(t) y = y(t) a ≤ t ≤ b Equivalently, C can be given by the vector equation r(t) = x(t) i + y(t) j. We assume that C is a smooth curve. This means that r′ is continuous and r′(t) ≠ 0. Dr. Erickson 16.2 Line Integrals

Definition If f is defined on a smooth curve C given by Equations 1, the line integral of f along C is: if this limit exists. Then, this formula can be used to evaluate the line integral. Dr. Erickson 16.2 Line Integrals

Example 1 – pg. 1096 #2 Evaluate the line integral, where C is the given curve. Dr. Erickson 16.2 Line Integrals

Line Integrals in Space We now suppose that C is a smooth space curve given by the parametric equations x = x(t) y = y(t) a ≤ t ≤ b or by a vector equation r(t) = x(t) i + y(t) j + z(t) k Dr. Erickson 16.2 Line Integrals

Line Integrals in Space Suppose f is a function of three variables that is continuous on some region containing C. Then, we define the line integral of f along C (with respect to arc length) in a manner similar to that for plane curves: We evaluate it using Dr. Erickson 16.2 Line Integrals

Example 2 – pg. 1096 #10 Evaluate the line integral, where C is the given curve. Dr. Erickson 16.2 Line Integrals

Example 3 Evaluate the line integral, where C is the given curve. Dr. Erickson 16.2 Line Integrals

Line Integrals of Vector Fields Definition - Let F be a continuous vector field defined on a smooth curve C given by a vector function r(t), a ≤ t ≤ b. Then, the line integral of F along C is: Dr. Erickson 16.2 Line Integrals

Notes When using Definition 13 on the previous slide, remember F(r(t)) is just an abbreviation for F(x(t), y(t), z(t)) So, we evaluate F(r(t)) simply by putting x = x(t), y = y(t), and z = z(t) in the expression for F(x, y, z). Notice also that we can formally write dr = r′(t) dt. Dr. Erickson 16.2 Line Integrals

Example 4 – pg. 1097 #20 Evaluate the line integral , where C is the given by the vector function r(t). Dr. Erickson 16.2 Line Integrals

Example 5 – pg. 1097 #22 Evaluate the line integral , where C is the given by the vector function r(t). Dr. Erickson 16.2 Line Integrals