Page 247 - Schaum's Outline of Theory and Problems of Advanced Calculus

238 LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS [CHAP. 10

Fig. 10-6

Solved Problems

LINE INTEGRALS
ð
ð1;2Þ
2
2
10.1. Evaluate ðx yÞ dx þðy þ xÞ dy along (a)a straight line from ð0; 1Þ to ð1; 2Þ,(b) straight
ð0;1Þ 2
lines from ð0; 1Þ to ð1; 1Þ and then from ð1; 1Þ to ð1; 2Þ,(c) the parabola x ¼ t, y ¼ t þ 1.
(a)An equation for the line joining ð0; 1Þ and ð1; 2Þ in the xy plane is y ¼ x þ 1. Then dy ¼ dx and the line
integral equals
ð 1 ð 1
2 2 2
fx ðx þ 1Þg dx þfðx þ 1Þ þ xg dx ¼ ð2x þ 2xÞ dx ¼ 5=3
x¼0 0
(b)Along the straight line from ð0; 1Þ to ð1; 1Þ, y ¼ 1; dy ¼ 0 and the line integral equals
ð 1 ð 1
2 2
ðx 1Þ dx þð1 þ xÞð0Þ¼ ðx 1Þ dx ¼ 2=3
x¼0 0
Along the straight line from ð1; 1Þ to ð1; 2Þ, x ¼ 1; dx ¼ 0 and the line integral equals
2 2
ð ð
2 2
ð1 yÞð0Þþð y þ 1Þ dy ¼ ð y þ 1Þ dy ¼ 10=3
y¼1 1
Then the required value ¼ 2=3 þ 10=3 ¼ 8:3.
(c) Since t ¼ 0at ð0; 1Þ and t ¼ 1at ð1; 2Þ,the line integral equals
ð 1 ð 1
2 2 2 2 5 3 2
ft ðt þ 1Þg dt þfðt þ 1Þ þ tg 2tdt ¼ ð2t þ 4t þ 2t þ 2t 1Þ dt ¼ 2
t¼0 0

ð
2
2
10.2. If A ¼ð3x 6yzÞi þð2y þ 3xzÞj þð1 4xyz Þk, evaluate A dr from ð0; 0; 0Þ to ð1; 1; 1Þ along
the following paths C: C
2
ðaÞ x ¼ t; y ¼ t ; z ¼ t 3
ðbÞ The straight lines from ð0; 0; 0Þ to ð0; 0; 1Þ, then to ð0; 1; 1Þ, and then to ð1; 1; 1Þ
ðcÞ The straight line joining ð0; 0; 0Þ and ð1; 1; 1Þ
ð ð
2 2
A dr ¼ fð3x 6yzÞi þð2y þ 3xzÞj þð1 4xyz Þkg ðdxi þ dyj þ dzkÞ
C C
ð
2
2
ð3x 6yzÞ dx þð2y þ 3xzÞ dy þð1 4xyz Þ dz
¼
C
3
2
(a)If x ¼ t; y ¼ t ; z ¼ t , points ð0; 0; 0Þ and ð1; 1; 1Þ correspond to t ¼ 0 and t ¼ 1, respectively. Then