Page 109 - Schaum's Outline of Theory and Problems of Advanced Calculus

100 INTEGRALS [CHAP. 5

n
X 2 2 1=2
lim fð x k Þ þð y k Þ g
k¼1
n!1
or equivalently
( ) 1=2
n 2
X y k
lim 1 þ ð x k Þ
k¼1
n!1 x k
where x k ¼ x k x k 1 and y k ¼ y k y k 1 .
Thus, the length of the arc of a curve in rectangular Cartesian coordinates is
( ) 1=2
b dx dy
ð ð 2 2
2 2 1=2
0 0 dt
L ¼ f½ f ðtÞ Šþ½g ðtފ g dt ¼ þ
a dt dt
(This form may be generalized to any number of dimensions.)
Upon changing the variable of integration from t to x we obtain the planar form
( ) 1=2
ð 2
dy
f ðbÞ
L ¼ 1 þ
dx
f ðaÞ
(This form is only appropriate in the plane.)
2
2
2
The generic differential formula ds ¼ dx þ dy is useful, in that various representations algebrai-
cally arise from it. For example,
ds
dt
expresses instantaneous speed.
AREA
Area was a motivating concept in introducing the integral. Since many applications of the integral
are geometrically interpretable in the context of area, an extended formula is listed and illustrated below.
Let f and g be continuous functions whose graphs intersect at the graphical points corresponding to
x ¼ a and x ¼ b, a < b.If gðxÞ A f ðxÞ on ½a; bŠ, then the area bounded by f ðxÞ and gðxÞ is

ð b
fgðxÞ f ðxÞg dx
A ¼
a
If the functions intersect in ða; bÞ, then the integral yields an algebraic sum. For example, if
gðxÞ¼ sin x and f ðxÞ¼ 0 then:
ð 2 2

sin xdx ¼ cos x ¼ 0

0 0

VOLUMES OF REVOLUTION
Disk Method
Assume that f is continuous on a closed interval a @ x @ b and that f ðxÞ A 0. Then the solid
realized through the revolution of a plane region R (bound by f ðxÞ, the x-axis, and x ¼ a and x ¼ b)
about the x-axis has the volume
ð b
2
V ¼ ½ f ðxފ dx
a