Page 391 - Advanced engineering mathematics

12.1 Line Integrals 371

C 4
z

C 3
y

C
C 1 2
x
FIGURE 12.2 A piecewise
smooth curve.

Write

C = C 1 C 2 ··· C n
if, as in Figure 12.2, C begins with a smooth piece C 1 . C 2 begins where C 1 ends, C 3 begins where
C 2 ends, and so on. Each C j is smooth, but where C j joins with C j+1 there may be no tangent in
the resulting curve. For C formed in this way,
4.

fdx + gdy + hdz = fdx + gdy + hdz
C C 1 C 2 C n
···
n

= fdx + gdy + hdz.
j=1 C j

EXAMPLE 12.5
2
2
Let C be the curve consisting of the quarter circle x + y = 1inthe x, y - plane from
(1,0) to (0,1), followed by the horizontal line segment from (0,1) to (2,1). We will compute
2
dx + y dy.
C
Write C = C 1 ⊕ C 2 , where C 1 is the quarter circle part and C 2 the line segment part.
Parametrize C 1 by x = cos(t), y = sin(t) for 0 ≤ t ≤ π/2. On C 1 ,
dx =−sin(t)dt and dy = cos(t)dt,
so
π/2 2

2 2
dx + y dy = [−sin(t) + sin (t)cos(t)]dt =− .
3
C 1 0
Parametrize C 2 by x = s, y = 1for0 ≤ s ≤ 2. On C 2 ,
dx = ds and dy = 0
so
2
2
dx + y dy = ds = 2.
C 2 0
Then
2 4
dx + y dy =− + 2 = .
2
3 3
C

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October 14, 2010 14:53 THM/NEIL Page-371 27410_12_ch12_p367-424