Line Integrals

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Section 17.2

• Line Integrals

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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 ti - 1,
tiof 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 ?si. 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.
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EVALUATING LINE INTEGRALS
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.
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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.
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EXAMPLE
Evaluate the following line integral where C
is the line segment joining (1, 2) to (4, 7)
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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,
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EXAMPLE
Evaluate where C is the piecewise-
smooth curve formed by the boundary region
bounded by y x and y x2.
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AN INTERPRETATION OF THE LINE INTEGRAL
Suppose that ?(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
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EXAMPLE
A thin wire is bent in the shape of the
semicircle x cos t, y sin t, 0 t
p 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.
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LINE INTEGRALS WITH RESPECT TO x AND y
Two other line integrals can be obtained by
replacing ?si by either ?xi xi - xi - 1 or
?yi yi - yi - 1. They are called the line
integrals of f along C with respect to x and y.
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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.
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EVALUATING LINE INTEGRALS WITH RESPECT TO x AND y
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A SPECIAL NOTATION
The line integrals with respect to x and y
frequently occur together. We write this as
follows.
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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
NOTE The line integral with respect to arc
length DOES NOT change sign.
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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
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EVALUATING LINE INTEGRALS IN SPACE
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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
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WORK AND LINE INTEGRALS
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 ?si by dividing the parameter
interval a, b into subintervals of equal width.
Choose a point on the ith
subarc corresponding to the parameter . If
?si 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
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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,
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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
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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, 3p)
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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.
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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.
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NOTE 2
where F Pi Qj Rk