Multivariate Calculus - Bruce E. Shapiro

LECTURE 22. LINE INTEGRALS 183
so that the second integral becomes
∫
C 2
F · dr =
=
=
=
∫ 1
0
∫ 1
0
∫ 1
0
(x, 2y) · d
dt
(1 − t, 2t)dt
(1 − t, 4t) · (−1, 2)dt
(−1 + 9t)dt
(−t + 9t2
2
)∣ ∣∣∣
1
0
= 7 2
Therefore the integral over C is
∮ ∫ ∫
F · dr = F · dr + F · dr = − 7
C
C 1 C 2
2 + 7 2 = 0
Example 22.4 A particle travels along the helix
in the vector field
r = (cos t, sin t, 2t)
F = (x, z, −xy)
Find the total work done over the time period 0 ≤ t ≤ 3π.
Solution. The work is given by
W =
=
=
=
∫ 3π
0
∫ 3π
0
∫ 3π
0
∫ 3π
0
= −3
F · r ′ dt
(x, z, −xy) · d (cos t, sin t, 2t)dt
dt
(cos t, 2t, − cos t sin t) · (− sin t, cos t, 2)dt
(− cos t sin t + 2t cos t − 2 cos t sin t) dt
∫ 3π
0
∫ 3π
cos t sin tdt + 2 t cos tdt
0
Using the integral formulas ∫ sin t cos tdt = 1 2 sin2 t and ∫ t cos t = cos t + t sin t gives
( )∣ 1 ∣∣∣
3π
W = −3
2 sin2 t + 2 (cos t + t sin t)| 3π
0
0
= 2(cos 3π − cos 0) = −4
Math 250, Fall 2006 Revised December 6, 2006.