Section 16.2: Line Integrals
Goals:
1. To write and evaluate line integrals
2. To write and evaluate a line integral of a vector field
3. To write and evaluate a line integral in differential form
We know how to integrate over an interval on the x-axis, a region in the xy-plane, and even a solid in
space. In this section we investigate how to integrate a function f ( x, y, z ) over a curve C. In fact, the
word "line" in the title of this section is not necessarily referring to a straight line. It refers to the curve C.
1. A vector-valued function r(t) is smooth for all t in an interval I = [a, b] if all of its
component functions have continuous derivatives on I and r′(t ) ≠ 0 for any t in (a, b) .
Recall:
t
2. The arc length function is defined as given by s(t) =
∫ r ′(u) du .
So, from the
a
ds
Fundamental Theorem of Calculus ,
= r′(t ) (see section 14.3).
dt
Definition:
A curve C given by the parameterization r(t) on the interval[a, b] is piecewise smooth if the
interval [a, b] can be divided up into a finite number of subintervals so that C is smooth on
each subinterval. A piecewise smooth curve is called a path
We assume that f ( x, y, z ) is a continuous function with domain that includes the smooth curve C given
by r (t ) = x(t )i + y (t ) j + z (t )k , a ≤ t ≤ b . Divide up the curve into a finite number of subarcs, and call the
length of the ith subarc ∆si . Now, let ( xi , yi , zi ) be a point chosen from the ith subarc and form the sum
n
∑ f ( x , y , z )∆s
i
i =1
i
i
i
. As n increases and the lengths of ∆si approach zero, it can be proved that the
n
sums, ∑ f ( xi , yi , zi )∆si , approach a limit. This limit is called the line integral of f over C, denoted
i =1
∫ f ( x, y, z )ds .
We evaluate
C
∫ f ( x, y, z )ds using the result ds =
r′(t ) dt = [ x′(t )]2 + [ y′(t )]2 + [ z ′(t )]2 dt .
C
Theorem 1: If C is a smooth curve given by r (t ) = x(t )i + y (t ) j + z (t )k , where a ≤ t ≤ b , and f is a
continuous function with a domain that includes C, then
b
∫ f ( x, y, z )ds = ∫ f ( x(t ), y(t ), z(t ))
C
Notes:
1.
2.
3.
4.
5.
[ x′(t )]2 + [ y′(t )]2 + [ z ′(t )]2 dt
a
If f ( x, y, z ) = 1 the line integral gives the arc length of C.
A similar formula holds for two-variable functions (see page 1099).
This integral is sometimes called a line integral with respect to arc length.
In order to evaluate a line integral, we must find a parameterization for C. This may
involve dividing up the interval [a, b] into subintervals so that C is smooth on each
subinterval (piecewise smooth)and then using the additive property of integrals to get
the answer.
The value of a line integral is the same no matter how C is parameterized (as long as it
is piecewise smooth).
Suppose we have a continuous vector field that represents some force in space, and an object is moving in
the vector field along the curve given by r(t) where a ≤ t ≤ b . Our goal is to calculate the work done by
the vector field in moving this object along the curve. We already know that work equals force times
distance if the path is a straight line and the force is constant in the direction of the line of motion. In
chapter 13 we generalized this formula to W = F • D where F is a constant force and D is the vector from
the beginning to the end of the line of movement. Now we're in the situation where the curve is not a
straight line and the force is not constant. So we partition the curve into subarcs that are so short that they
are nearly straight lines and the force is close to being constant. Hopefully, you can see that we're headed
towards a line integral- definition of work, where D becomes dr and the work performed over the subarc
is given by dW= F • dr . So the total work performed is given by ∫ F • dr . Now, dr = r′(t )dt . So, the
C
b
formula for work becomes ∫ F • dr = ∫ F( x(t ), y (t ), z (t )) • r′(t )dt .
C
Note:
a
This integral appears in other applications so a more general name is given to it: The line
b
integral of a vector field F is given by ∫ F • dr = ∫ F( x(t ), y (t ), z (t )) • r′(t )dt .
C
Note:
It turns out that
∫ F • dr = − ∫ F • dr (see page 1106 for details).
−C
Definition:
a
C
Let F ( x, y ) = P( x, y )i + Q ( x, y ) j be a vector field. A line integral over this vector field,
∫ F • dr ,
can be rewritten in the differential form
C
∫ ( Pdx + Qdy)
C
See page 1107 for the three variable version of differential form.