Multivariate Calculus - Bruce E. Shapiro

182 LECTURE 22. LINE INTEGRALS
Thus the circulation falls off inversely as we move away from the origin, approaching
zero as the arrows become more and more parallel, as our intuitive notion suggested.
Example 22.3 Compute the line integral of the function F = (x, 2y) over the curve
C shown in figure 22.5.
Figure 22.5: Integration path for example 22.3.
Solution. To complete this integral, we divide the curve into two parts. Let us call
the parabola y = 2−2x 2 from (0, 2) to (1, 0) by the name C 1 , and let us call the line
back along y = 2 − 2x from (1, 0) to (0, 2) by the name C 2 , as indicated in figure
22.5. ∮
F · dr =
C
∫
F · dr +
C 1
∫
F · dr
C 2
We can parameterize the path C 1 by
x = t, y = 2 − 2t 2 , 0 ≤ t ≤ 1
Thus on C 1 , r = (t, 2 − 2t 2 ), r ′ = (1, −4t), and F = (x, 2y) = (t, 4 − 4t 2 ) and
∫ ∫ 1
F · dr = (t, 4 − 4t 2 ) · (1, −4t)dt
C 1 0
=
∫ 1
0
(t − 16t + 16t 3 )dt
)∣
=
(− 15t2
2 + 16t4 ∣∣∣
1
4
0
= − 15
2 + 4 = −7 2
On the line y = 2 − 2x, we start at the point (1,0) and move toward the point (0,
2). We can parameterize this as
x = 1 − t, y = 2 − 2(1 − t) = 2t, 0 ≤ t ≤ 1
Revised December 6, 2006. Math 250, Fall 2006