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.)