me 2304: 3d geometry & vector calculus dr. faraz junejo line integrals
TRANSCRIPT
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ME 2304: 3D Geometry & Vector Calculus
Dr. Faraz Junejo
Line Integrals
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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
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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.
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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 ),(
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• 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
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• 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
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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.
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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.
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• 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
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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
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• 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
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• Just as for an ordinary single integral, we can
interpret the line integral of a positive
function as an area.
Line Integrals
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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
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Example: 1
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Example: 1 (contd.)
tu
duu
tdt
dt tNote
55
4
4
sin5
1
5
cosdu
sint u Let cossin that
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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
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• Also, the upper half of the circle is described
by the parameter interval 0 ≤ t ≤ π
Exercise: 1 (contd.)
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• 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.)
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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
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Piecewise smooth Curves
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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,
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Example: 2
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• At first we need to parameterize each of the
curves, i.e.
Example: 2 (contd.)
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Example: 2 (contd.)
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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.
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• 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.)
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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
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Example: 3 (contd.)
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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.
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Example: 4
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• 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.)
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• 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.)
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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.
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• 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.)
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• 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.)
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• 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
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• 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.)
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• 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
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• 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
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• Thus,
1 2
2 2 2
5 5 12
6
C C Cx ds x ds x ds
Exercise: 1 (Contd.)