Page 249 - Schaum's Outline of Theory and Problems of Advanced Calculus

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

In traversing C we have chosen the counterclockwise direction indicated in Fig. 10-7. We call this the
positive direction, or say that C has been traversed in the positive sense. If C were tranversed in the
clockwise (negative) direction, the value of the integral would be 96 .

ð
ffiffiffi
10.4. Evaluate yds along the curve C given by y ¼ 2 x from x ¼ 3to x ¼ 24.
p
C
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 0 2 p 1 þ 1=x dx,we have
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Since ds ¼ dx þ dy ¼ 1 þð y Þ dx ¼
24
ð ð ð 24
ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip 24 p ffiffiffiffiffiffiffiffiffiffiffi 4
p
2 x 1 þ 1=x dx ¼ 2 3=2 ¼ 156
yds ¼ x þ 1 dx ¼ ðx þ 1Þ
C 2 3 3 3
GREEN’S THEOREM IN THE PLANE
10.5. Prove Green’s theorem in the plane if C is a closed curve
which has the property that any straight line parallel to
the coordinate axes cuts C in at most two points.
Let the equations of the curves AEB and AFB (see adjoin-
ing Fig. 10-8) be y ¼ Y 1 ðxÞ and y ¼ Y 2 ðxÞ, respectively. If r is
the region bounded by C,we have
ðð ð b ð
@P Y 2 ðxÞ @P
dy dx
@y x¼a y¼Y 1 ðxÞ @y Fig. 10-8
dx dy ¼
r
ð b ð b
½Pðx; Y 2 Þ Pðx; Y 1 ފ dx
Y 2 ðxÞ
¼ Pðx; yÞj dx ¼
y¼Y 1 ðxÞ
x¼a a
ð b ð a þ
Pdx
¼ Pðx; Y 1 Þ dx Pðx; Y 2 Þ dx ¼
a b C
Then
þ ðð
@P
dx dy
ð1Þ Pdx ¼ @y
C
r
Similarly let the equations of curves EAF and EBF be x ¼ X 1 ð yÞ and x ¼ X 2 ð yÞ respectively. Then
ðð ð f ð ð f
@Q X 2 ð yÞ @Q
½QðX 2 ; yÞ QðX 1 ; yފ dy
@x dx dy ¼ y¼c x¼x 1 ð yÞ @x dx dy ¼ c
r
ð c ð f þ
Qdy
¼ QðX 1 ; yÞ dy þ QðX 2 ; yÞ dy ¼
f c C
@Q
þ ðð
Then ð2Þ Qdy ¼ dx dy
C @x
r
þ ðð
@Q @P
Adding (1) and (2), Pdx þ Qdy ¼ dx dy
C @x @y
r
10.6. Verify Green’s theorem in the plane for
þ
2
2
ð2xy x Þ dx þðx þ y Þ dy
C
2 2
where C is the closed curve of the region bounded by y ¼ x and y ¼ x.