ME 2304: 3D Geometry & Vector Calculus
Dr. Faraz Junejo
Line Integrals
In this lecture, we define an integral that
is similar to a single integral except that, instead
of integrating over an interval [a, b], we
integrate over a curve C.
– Such integrals are called line integrals.
Line Integral
Line Integral
• In mathematics, a line integral (sometimes
called a path integral, contour integral, or
curve integral) is an integral where the
function to be integrated is evaluated along a
curve.
• The function to be integrated may be a scalar
field or a vector field.
Line Integrals (Contd.)Consider the following problem:
• A piece of string, corresponding to a curve C, lies in
the xy-plane. The mass per unit length of the string is
f(x,y). What is the total mass of the string?
• The formula for the mass is:
• The integral above is called a line integral of f(x,y)
along C.
C
dsyxfMass ),(
• We use a ds here to acknowledge the fact that
we are moving along the curve, C, instead of
the x-axis (denoted by dx) or the y-axis
(denoted by dy).
• Because of the ds this is sometimes called the
line integral of f with respect to arc length.
Line Integrals with Respect to Arc Length
• Question: how do we actually evaluate the above integral?
• The strategy is:
(1) parameterize the curve C,
(2) cut up the curve C into infinitesimal pieces i.e.
small pieces,
(3) determine the mass of each infinitesimal piece,
(4) integrate to determine the total mass.
Line Integrals with Respect to Arc Length
Arc Length• We’ve seen the notation ds before. If you recall from
Calculus I course, when we looked at the arc length of
a curve given by parametric equations we found it to
be,
• It is no coincidence that we use ds for both of these
problems. The ds is the same for both the arc length
integral and the notation for the line integral.
Computing Line Integral
• So, to compute a line integral we will convert
everything over to the parametric equations.
The line integral is then,
• Don’t forget to plug the parametric equations
into the function as well.
• If we use the vector form of the
parameterization we can simplify the notation
up by noticing that,
• Using this notation the line integral becomes,
Computing Line Integral
Special Case
• In the special case where C is the line
segment that joins (a, 0) to (b, 0), using x as
the parameter, we can write the parametric
equations of C as:
• x = x
• y = 0
• a ≤ x ≤ b
• Line Integral formula then becomes
– So, the line integral reduces to an ordinary
single integral in this case.
, ,0b
C af x y ds f x dx
Special Case
• Just as for an ordinary single integral, we can
interpret the line integral of a positive
function as an area.
Line Integrals
Line Integrals• In fact, if f(x, y) ≥ 0, represents
the area of one side of the “fence” or “curtain” shown here, whose:
– Base is C.
– Height above the point (x, y) is f(x, y).
,C
f x y ds
Example: 1
Example: 1 (contd.)
tu
duu
tdt
dt tNote
55
4
4
sin5
1
5
cosdu
sint u Let cossin that
Exercise: 1• Evaluate
where C is the upper half of the unit circle x2 + y2 = 1
– To use Line Integral Formula, we first need parametric
equations to represent C.
– Recall that the unit circle can be parametrized by
means of the equations
x = cos t y = sin t
22C
x y ds
• Also, the upper half of the circle is described
by the parameter interval 0 ≤ t ≤ π
Exercise: 1 (contd.)
• So, using Line integral Formula gives:
2 22 2
0
2 2 2
0
2
0
323
0
2 2 cos sin
2 cos sin sin cos
2 cos sin
cos2 2
3
C
dx dyx y ds t t dt
dt dt
t t t t dt
t t dt
tt
Exercise: 1 (contd.)
Exercise: 2• Evaluate
Where, C is the upper right quarter of a circle x2 + y2 = 16, rotated in counterclockwise
direction.
dsxyC 2
Answer: 256/3
Piecewise smooth Curves
Piecewise smooth Curves• Evaluation of line integrals over piecewise
smooth curves is a relatively simple thing to
do. All we do:
• is evaluate the line integral over each of the pieces
and then add them up.
• The line integral for some function over the above
piecewise curve would be,
Example: 2
• At first we need to parameterize each of the
curves, i.e.
Example: 2 (contd.)
Example: 2 (contd.)
Example: 2 (contd.)
Notice that we put direction arrows on the curve in this example.
The direction of motion along a curve may change the value of the line integral as we will see in the next example.
• Also note that the curve in this example can be
thought of a curve that takes us from the point
(-2,-1) to the point (1, 2) .
• Let’s first see what happens to the line integral
if we change the path between these two
points.
Example: 2 (contd.)
Example: 3
vector form of the equation of a line
we know that the line segment start at (-2,-1) and ending at (1, 2) is given by,
3,3(-2,-1)-(1,2)ba,
be lvector wildirection & 1,2;
),(),(
o
ooo
rhere
batyxvtrr
Example: 3 (contd.)
Summary: Example: 2 & 3
• So, the previous two examples seem to suggest
that if we change the path between two points
then the value of the line integral (with respect
to arc length) will change.
• While this will happen fairly regularly we can’t
assume that it will always happen. In a later
section we will investigate this idea in more detail
• Next, let’s see what happens if we change the
direction of a path.
Example: 4
• So, it looks like when we switch the direction
of the curve the line integral (with respect to
arc length) will not change.
• This will always be true for these kinds of line
integrals.
• However, there are other kinds of line
integrals (discussed in Exercise: 2 later on) in
which this won’t be the case.
Example: 4 (contd.)
• We will see more examples of this in next
sections so don’t get it into your head that
changing the direction will never change the
value of the line integral.
Example: 4 (contd.)
Fact: Curve Orientation
• Let’s suppose that the curve C has the parameterization x = h(t ) , y = g (t )
• Let’s also suppose that the initial point on the curve is A and the final point on the curve is B.
• The parameterization x = h(t ) , y = g (t )
will then determine an orientation for the curve where
the positive direction is the direction that is traced (i.e.
drawn) out as t increases.
• Finally, let -C be the curve with the same
points as C, however in this case the curve has
B as the initial point and A as the final point.
• Again t is increasing as we traverse this curve.
In other words, given a curve C, the curve -C is
the same curve as C except the direction has
been reversed.
Fact (Contd.)
• For instance, here– The initial point A
corresponds to
the parameter value.
– The terminal point B
corresponds to t = b.
– We then have the following fact about line integrals with
respect to arc length.
Fact (Contd.)
• Evaluate
• where C consists of the arc C1 of the parabola
y = x2 from (0, 0) to (1, 1) followed by the
vertical line segment C2 from (1, 1) to (1, 2).
2C
x ds
Exercise: 1
• The curve is shown here.
• C1 is the graph of a function of x, as y = x2
– So, we can choose t as the parameter.
– Then, the equations for C1 become:
x = t y = t2 0 ≤ t ≤ 1
Exercise: 1 (Contd.)
• Therefore,Exercise: 1 (Contd.)
7.16
15541
3
2
4
1
4
1
4/12
841ulet
412
22
1
0
2/32
1
0
2/1
2
1
0
2
1
1
0
22
t
duu
dutdt
tdtdutNow
dttt
dtdt
dy
dt
dxtxds
C
• On C2, we choose y as the parameter.– So, the equations of C2
are: x = 1 y = t 1 ≤ t ≤ 2 and
Exercise: 1 (Contd.)
2102
122
1
0
2
1
0
22
dtt
dtdt
dy
dt
dxxds
C
• Thus,
1 2
2 2 2
5 5 12
6
C C Cx ds x ds x ds
Exercise: 1 (Contd.)