Chapter 16 – Vector Calculus 16.2 Line Integrals 1 Objectives: Understand various aspects of line...
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Transcript of Chapter 16 – Vector Calculus 16.2 Line Integrals 1 Objectives: Understand various aspects of line...
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Chapter 16 – Vector Calculus16.2 Line Integrals
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 IntegralsThey were invented in the early 19th century to solve
problems involving:
◦ Fluid flow
◦ Forces
◦ Electricity
◦Magnetism
Dr. Erickson
16.2 Line Integrals 3
Line IntegralsWe 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.
* *
1
, lim ,n
i i iC ni
f x y ds f x y s
2 2
, ,b
C a
dx dyf x y ds f x t y t dt
dt dt
Dr. Erickson
16.2 Line Integrals 5
Example 1 – pg. 1096 #2Evaluate the line integral, where C is the given curve.
2, : , 2 , 0 1C
xy ds C x t y t t
Dr. Erickson
16.2 Line Integrals 6
Line Integrals in SpaceWe 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 SpaceSuppose 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
* * *
1
, , lim , ,n
i i i iC ni
f x y z ds f x y z s
2 2 2
, , , ,b
C a
dx dy dzf x y z ds f x t y t z t
dt dt dt
Dr. Erickson
16.2 Line Integrals 8
Example 2 – pg. 1096 #10Evaluate the line integral, where C is the given curve.
2 ,
is the line segment from 1,5,0 to 1,6,4C
xyz ds
C
Dr. Erickson
16.2 Line Integrals 9
Example 3Evaluate the line integral, where C is the given curve.
2 2 2 ,
consists of the line segments from 0,0,0
to 1,2, 1 and from 1,2, 1 to 3, 2,0 .
C
x dx y dy z dz
C
Dr. Erickson
16.2 Line Integrals 10
Line Integrals of Vector FieldsDefinition - 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:
'b
C a Cd t t dt ds F r F r r F T
Dr. Erickson
16.2 Line Integrals 11
NotesWhen 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. 1097 #20Evaluate the line integral , where C is the given by
the vector function r(t).
2
2 3 2
( , , ) ( ) ( ) ,
( ) , 0 1
x y z x y y z z
t t t t t
F i j k
r i j k
CdF r
Dr. Erickson
16.2 Line Integrals 13
Example 5 – pg. 1097 #22Evaluate the line integral , where C is the given by
the vector function r(t).
( , , ) ,
( ) cos sin , 0
x y z x y xy
t t t t t
F i j k
r i j k
CdF r
Dr. Erickson
