Page 248 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 10] LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS 239

ð ð 1
2 2 3 2 3 2 2 3 2 3
A dr ¼ f3t 6ðt Þðt Þg dt þf2t þ 3ðtÞðt Þg dðt Þþf1 4ðtÞðt Þðt Þ g dðt Þ
C t¼0
ð 1
5
2
3
11
2
5
ð3t 6t Þ dt þð4t þ 6t Þ dt þð3t 12t Þ dt ¼ 2
¼
t¼0
Another method:
2
4
2
3
5
9
2
Along C, A ¼ð3t 6t Þi þð2t þ 3t Þj þð1 4t Þk and r ¼ xi þ yj þ zk ¼ ti þ t j þ t k,
2
dr ¼ði þ 2tj þ 3t kÞ dt. Then
ð ð 1
2
3
5
5
2
11
ð3t 6t Þ dt þð4t þ 6t Þ dt þð3t 12t Þ dt ¼ 2
A dr ¼
C 0
(b)Along the straight line from ð0; 0; 0Þ to ð0; 1; 1Þ, x ¼ 0; y ¼ 0; dx ¼ 0; dy ¼ 0, while z varies from 0 to 1.
Then the integral over this part of the path is
ð 1 ð 1
2 2
f3ð0Þ 6ð0ÞðzÞg0 þf2ð0Þþ 3ð0ÞðzÞg0 þf1 4ð0Þð0Þðz Þg dz ¼ dz ¼ 1
z¼0 z¼0
Along the straight line from ð0; 0; 1Þ to ð0; 1; 1Þ, x ¼ 0; z ¼ 1; dx ¼ 0; dz ¼ 0, while y varies from 0
to 1. Then the integral over this part of the path is
ð 1 ð 1
2 2
f3ð0Þ 6ð yÞð1Þg0 þf2y þ 3ð0Þð1Þg dy þf1 4ð0Þð yÞð1Þ g0 ¼ 2ydy ¼ 1
y¼0 y¼0
Along the straight line from ð0; 1; 1Þ to ð1; 1; 1Þ, y ¼ 1; z ¼ 1; dy ¼ 0; dz ¼ 0, while x varies from 0
to 1. Then the integral over this part of the path is
1 1
ð ð
2 2 2
f3x 6ð1Þð1Þg dx þf2ð1Þþ 3xð1Þg0 þf1 4xð1Þð1Þ g0 ¼ ð3x 6Þ dx ¼ 5
x¼0 x¼0
ð
Adding, A dr ¼ 1 þ 1 5 ¼ 3:
C
(c) The straight line joining ð0; 0; 0Þ and ð1; 1; 1Þ is given in parametric form by x ¼ t; y ¼ t; z ¼ t. Then
ð ð 1
2
4
2
2
ð3t 6t Þ dt þð2t þ 3t Þ dt þð1 4t Þ dt ¼ 6=5
A dr ¼
C t¼0
10.3. Find the work done in moving a particle once around an y
ellipse C in the xy plane, if the ellipse has center at the r = xi + yj
origin with semi-major and semi-minor axes 4 and 3, = 4 cos t i + 3 sin t j
respectively, as indicated in Fig. 10-7, and if the force r
field is given by t x
2
3
2
F ¼ð3x 4y þ 2zÞi þð4x þ 2y 3z Þj þð2xz 4y þ z Þk
2
In the plane z ¼ 0; F ¼ð3x 4yÞi þð4x þ 2yÞj 4y k and
dr ¼ dxi þ dyj so that the work done is Fig. 10-7
þ ð
2
F dr ¼ fð3x 4yÞi þð4x þ 2yÞj 4y kg ðdxi þ dyjÞ
C C
þ
ð3x 4yÞ dx þð4x þ 2yÞ dy
¼
C
Choose the parametric equations of the ellipse as x ¼ 4cos t, y ¼ 3 sin t, where t varies from 0 to 2 (see
Fig. 10-7). Then the line integral equals
ð 2
f3ð4cos tÞ 4ð3 sin tÞgf 4 sin tg dt þf4ð4cos tÞþ 2ð3 sin tÞgf3cos tg dt
t¼0
ð 2
2
2
ð48 30 sin t cos tÞ dt ¼ð48t 15 sin tÞj 0 ¼ 96
¼
t¼0