1. 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

2. 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

3. 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

4. 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

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

6. 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

7. 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

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

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

10. 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

11. 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

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

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