Line Integrals of Scalar Fields

Math 55 - Elementary Analysis III

Institute of MathematicsUniversity of the Philippines

Diliman

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Curtain Area Problem

Problem

Let C be a smooth curve defined by a vector function~R(t) = x(t) + y(t), with t [a, b] and suppose f(x, y) iscontinuous on C. Find the area of the curtain whose base is aportion of C and whose height at a point (x, y) on C is f(x, y).

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Solving the Curtain Area Problem

1 Partition [a, b] into n sub-intervals[ti1, ti] such that the correspondingarcs in C has equal lengths s.

2 Choose ti [ti1, ti]. This correspondsto a point P i (x

i , yi ) in each subarc.

3 Construct cylinders with each subarc as base andheight f(xi , y

i ).

4 Each cylinder will have area A = f(xi , yi )s and the

area of the curtain is approximately

A ni=1

A =

ni=1

f(xi , yi )s.

5 As n, the error vanishes and the area A of the curtain is

A = limn

ni=1

f(xi , yi )s.

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Line Integrals with respect to Arclength

Definition

Let f(x, y) be continuous on a smooth plane curve C defined bya vector function ~R(t) = x(t), y(t), t [a, b]. The lineintegral of f along C (with respect to the arclengthparameter) is

Cf(x, y) ds = lim

n

ni=1

f(xi , yi )s

provided this limit exists.

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Line Integrals with respect to Arclength

Remarks.

1 The line integral is independent of the parametrization ofC.

2 If C denotes the curve C traced in the opposite direction,then

Cf(x, y) ds =

C

f(x, y) ds

3 If C is piecewise smooth, i.e, C is the union of finite number ofsmooth curves C1, C2, . . . , Cn, thenC

f(x, y) ds =

C1

f(x, y) ds+

C2

f(x, y) ds+ . . .+

Cn

f(x, y) ds

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Evaluating Line Integrals

Recall thatds

dt= ~R(t). Hence,

Definition Cf(x, y) ds =

baf(x(t), y(t)) ~R(t) dt

If ~R(t) = x(t), y(t), then ~R(t) =

[x(t)]2 + [y(t)]2 andtherefore,

DefinitionCf(x, y) ds =

baf(x(t), y(t))

[x(t)]2 + [y(t)]2 dt

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Some Examples

Example

Evaluate

C

2xy ds where C is the part unit circle in the first

quadrant.

Solution. C can be parametrized by ~R(t) = cos t, sin t,0 t pi2 . Therefore,

C2xy ds =

pi2

02 cos t sin t

( sin t)2 + (cos t)2 dt

=

pi2

02 cos t sin t dt

= sin2 t

pi20

= 1

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Some Examples

Example

Evaluate

Cx ds where C is the part of the parabola y = x2

from (0, 0) to (1, 1), followed by the line segments from (1, 1) to(0, 2) and from (0, 2) to (0, 0).

Solution. C is a piecewise smooth curve which can be wittenas the union of the curves C1, C2 and C3. Therefore,

Cx ds =

C1

x ds +

C2

x ds +

C3

x ds

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Some Examples

Solution(contd).C1 is the part of the parabola y = x

2 from (0, 0) to (1, 1).

The parametrization is ~R1(t) =t, t2

, 0 t 1. Hence,

C1

x ds =

10

t

1 + 4t2 dt =5

5 112

C2 is the line segment from (1, 1) to (0, 2). Aparametrization will be~R2(t) = (1 t) 1, 1+ t 0, 2 = 1 t, 1 + t, 0 t 1.Hence,

C2

x ds =

10

(1 t)1 + 1 dt =

2

2

C3 is the line segment from (0, 2) to (0, 0). Using the

parametrization ~R3(t) = 0, t where 0 t 2, we haveC3

x ds =

20

0

0 + 1 dt = 0

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Some Examples

Solution(contd). Finally, we haveCx ds =

C1

x ds +

C2

x ds +

C3

x ds

=5

5 112

+

2

2+ 0

=5

5 + 6

2 112

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Line Integrals with respect to x and y

Definition

Let f(x, y) be continuous on a smooth curve C given by~R(t) = x(t), y(t), a t b.

1

Cf(x, y) dx = lim

n

ni=1

f(xi , yi )x

2

Cf(x, y) dy = lim

n

ni=1

f(xi , yi )y

provided these limits exist.

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Evaluating Line Integrals

Equivalently, we have the following definition

Definition

1

Cf(x, y) dx =

baf(x(t), y(t))x(t) dt

2

Cf(x, y) dy =

baf(x(t), y(t)) y(t) dt

Oftentimes, the line integrals with respect to x and y occurtogether. In this case, we have the following

DefinitionCP (x, y) dx +

CQ(x, y) dy =

CP (x, y) dx + Q(x, y) dy

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Evaluating Line Integrals

Remarks.1 If C denotes the curve C traced in the opposite direction,

then C

f(x, y) dx = C

f(x, y) dxC

f(x, y) dy = C

f(x, y) dy

2 If C is piecewise smooth such that C = C1 C2 . . . Cnwhere each Ci is smooth for i = 1, 2, . . . , n, thenC

f(x, y) dx =

C1

f(x, y) dx+

C2

f(x, y) dx+ . . .+

Cn

f(x, y) dxC

f(x, y) dy =

C1

f(x, y) dy +

C2

f(x, y) dy + . . .+

Cn

f(x, y) dy

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Line Integrals with respect to x and y

Example

Let C be the curve given by ~R(t) =t2, 2 t, t [0, 1].

Evaluate

Cxy dx + (2x y) dy

Solution.C

xy dx + (2x y) dy = 10

t2(2 t)(2t) dt + [2t2 (2 t)](1) dt

=

10

2t4 + 4t3 2t2 t + 2 dt

= 25t5 + t4 2

3t3 1

2t2 + 2t

10

= 25

+ 1 23 1

2+ 2

=43

30

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Exercises

Evaluate the following line integrals over the given curves C.

1

Cy3 ds, C is given by ~R(t) =

t3, t

, t [0, 2].

2

Cxy4 ds, C is the right half of the circle x2 + y2 = 16.

3

Cxey dx, C is the arc of the curve x = ey from (1, 0) to

(e, 1).

4

C

sinx dx + cos y dy, where C is the top half of the circle

x2 + y2 = 1 from (1, 0) to (1, 0) and the line segment from(1, 0) to (1, 0).

5

Cxyz2 ds, C is the line segment from (1, 5, 0) to (1, 6, 4).

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References

1 Stewart, J., Calculus, Early Transcendentals, 6 ed., ThomsonBrooks/Cole, 2008

2 Dawkins, P., Calculus 3, online notes available athttp://tutorial.math.lamar.edu/

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