Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤...

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Section 17.2 Line Integrals

Transcript of Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤...

Page 1: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

Section 17.2

Line Integrals

Page 2: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

Let C be a smooth plane curve given by

x = x(t), y = y(t), a ≤ t ≤ b.

We divide the parameter interval [a, b] into n subintervals [ti − 1, ti]of equal width, and we let xi = x(ti) and yi = y(ti). Then the corresponding points Pi(xi, yi) divide C into n subarcs with lengths Δsi. Let be a point on the subarc Ci. If f is defined on a smooth curve C, then the line integral of f along C

if the limit exists.

LINE INTEGRALS

),( ***ii yxP

i

C

n

iiii

nsyxfdsyxf

1

** ),(lim),(

Page 3: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

Recall from Section 11.2 that the arc length of C is

If f is a continuous function, the limit on the previous slide always exists. The following formula can be used to evaluate the line integral.

EVALUATING LINE INTEGRALS

b

a

dtdt

dy

dt

dxL

22

C

b

a

dtdt

dy

dt

dxtytxfdsyxf

22

)(),(),(

Page 4: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

INTERPRETATION OF THELINE INTEGRAL

If z = f (x, y) ≥ 0 and C is a curve in the plane, then line integral

gives the area of the curved “curtain” below the surface and above C. See Figure 2 on page 1099.

Cdsyxf ),(

Page 5: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

EXAMPLE

Evaluate the following line integral where

C is the line segment joining (1, 2) to (4, 7)

C

x dsey

Page 6: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

PIECEWISE-SMOOTH CURVES AND LINE INTEGRALS

If C is a piecewise-smooth curve then C can be written as a finite union of smooth curves; that is,

C = C1 U C2 . . . U Cn

The line integral of f along C is defined as the sum of the line integrals of f along each of the smooth pieces of C; that is,

nC

CCC

dsyxf

dsyxfdsyxfdsyxf

),(

),(),(),(21

Page 7: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

Evaluate where C is the piecewise-

smooth curve formed by the boundary region bounded by y = x and y = x2.

Cdsx

EXAMPLE

Page 8: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

AN INTERPRETATION OF THE LINE INTEGRAL

Suppose that ρ(x, y) represents the density of a thin wire that is shaped like the plane curve C. The mass of the wire is given by

The center of mass of the wire is given by

CC

C

dsyxym

ydsyxxm

x

dsyxm

),(1

),(1

),(

Page 9: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

EXAMPLE

A thin wire is bent in the shape of the semicircle

x = cos t, y = sin t, 0 ≤ t ≤ π

If the density of the wire at a point is proportional to its distance from the x-axis, find the mass and center of mass of the wire.

Page 10: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

LINE INTEGRALS WITH RESPECT TO x AND y

Two other line integrals can be obtained by replacing Δsi by either Δxi = xi − xi − 1 or Δyi = yi − yi − 1. They are called the line integrals of f along C with respect to x and y.

n

iiii

nC

n

iiii

nC

yyxfdyyxf

xyxfdxyxf

1

**

1

**

),(lim),(

),(lim),(

Page 11: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

DISTINGUISHING FROM THE ORIGINAL LINE INTEGRAL

To distinguish the line integral with respect to x and y from the original line integral ∫C f (x, y) ds, we call ∫C f (x, y) ds the line integral with respect to arc length.

Page 12: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

EVALUATING LINE INTEGRALS WITH RESPECT TO x AND y

b

aC

b

aC

dttytytxfdyyxf

dttxtytxfdxyxf

)()(),(),(

)()(),(),(

Page 13: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

A SPECIAL NOTATIONThe line integrals with respect to x and y frequently occur together. We write this as follows.

CCC

dyyxQdxyxPdyyxQdxyxP ),(),(),(),(

Page 14: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

ORIENTATION AND LINE INTEGRALS

Recall that a given parametrization x = x(t), y = y(t), a ≤ t ≤ b, determines an orientation of a curve C. If we let −C denote the curve consisting of the same points as C but with opposite orientation, then we have:

CC

CCCC

dsyxfdsyxf

dyyxfdyyxfdxyxfdxyxf

),(),(

),(),(),(),(

NOTE: The line integral with respect to arc length DOES NOT change sign.

Page 15: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

LINE INTEGRALS IN SPACE

Suppose that C is a smooth space curve given by

x = x(t), y = y(t), z = z(t), a ≤ t ≤ b.

Suppose that f is function of three variables that is continuous on some region containing C, then the line integral of f along C is defined in a similar manner as for plane curves:

n

iiiii

nCszyxfdszyxf

1

*** ),,(lim),,(

Page 16: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

EVALUATING LINE INTEGRALS IN SPACE

b

aC

dtdt

dz

dt

dy

dt

dxzyxfdszyxf

222

),,(),,(

Page 17: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

VECTOR NOTATION FOR LINE INTEGRALS

If r(t) is the vector form of either a plane curve or a space curve, then the formula for evaluating a line integral with respect to arc length can be written compactly as

dtttfb

a|)(|)( rr

Page 18: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

Suppose that F = P i + Q j + R k is a continuous force field in three dimensions, such as a gravitational field. To compute the work done by this force in moving a particle along the smooth curve C, we divide C into subarcs Pi−1Pi with lengths Δsi by dividing the parameter interval [a, b] into subintervals of equal width. Choose a point on the ith subarc corresponding to the parameter . If Δsi is small, then as the particle moves from Pi−1 to Pi along the curve, it proceeds approximately in the direction of , the unit tangent vector at . The work done by the force F in moving to particle from Pi−1 to Pi is approximately

The total work done in moving the particle along C is approximately

WORK AND LINE INTEGRALS

),,( ****iiii zyxP

*it

)( *itT

*iP

iiiiiiiiii stzyxtszyx )(),,()(),,( ******** TFTF

n

iiiiii stzyx

1

**** )(),,( TF

Page 19: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

WORK (CONCLUDED)

Based on the derivation on the previous slide, we define the work W done by the force field F in moving a particle along C as the limit of the Riemann sums, namely,

CC

dsdszyxzyxW TFTF ),,(),,(

Page 20: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

EVALUATING A LINE INTEGRAL FOR WORK

If the curve C is given by the vector equation

r(t) = x(t)i + y(t)j + z(t)k

then T(t) = r′(t)/|r′(t)|. So, the line integral for work can be rewritten as

dtttdttt

ttW

b

a

b

a

)()(|)(||)(|

)()( rrFr

r

rrF

Page 21: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

EXAMPLE

Find the work done by the force field

on a particle as it moves along the helix given by

r(t) = cos ti + sin tj + tk

from the point (1, 0, 0) to (−1, 0, 3π)

kjiF4

1

2

1

2

1),,( yxzyx

Page 22: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

LINE INTEGRAL OF A VECTOR FIELD

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

where F = Pi + Qj + Rk.

C

b

aCdsdtttd TFrrFrF )()(

Page 23: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

NOTE 1

Even though ∫C F ∙ dr = ∫C F ∙ T ds and integrals with respect to arc length are unchanged when orientation is reversed, it is still true that

because the unit tangent vector T is replaced by its negative when C is replaced by −C.

CC

dd rFrF

Page 24: Section 17.2 Line Integrals. Let C be a smooth plane curve given by x = x(t), y = y(t), a ≤ t ≤ b. We divide the parameter interval [a, b] into n subintervals.

NOTE 2

CC

dzRdyQdxPdrF

where F = Pi + Qj + Rk