length Lbetween two points a= [x(ta), y (ta)] and b= [x(tb), y (tb)] on the curve is given
by the integral
L=
∫
C
ds =
∫tb
ta
√(
dx
dt
) 2
+
(
dy
dt
) 2
dt
If the curve Crepresents a wire with variable density f(x, y )[gm/cm], then the total
mass mof the wire between the points aand bis given by m=
∫
C
f(x, y )ds which
can be thought of as the limit of a summation process. If the curve Cis partitioned
into npieces of lengths ∆s 1 ,∆s 2 ,... ,∆si,.. ., then in the limit as nincreases without
bound and ∆siapproaches zero, one can express the total mass mof the wire as the
limiting process
m=
∫
C
f(x, y )ds = lim∆si→ 0
n→∞
∑n
i=1
f(x∗i, y∗i) ∆ si=
∫tb
ta
f(x(t), y (t))
√(
dx
dt
) 2
+
(
dy
dt
) 2
dt
where (x∗i, yi∗)is a general point on the ∆siarc length.
Whenever the values of xand yare restricted to lie on a given curve defined by
x=x(t)and y=y(t)for ta≤t≤tb, then integrals of the form
I=
∫
C
P(x, y )dx +Q(x, y )dy =
∫tb
ta
P(x(t), y (t))x′(t)dt +Q(x(t), y (t))y′(t)dt (12.186)
are called line integrals and are defined by a limiting process such as above. Line
integrals are reduced to ordinary integrals by substituting the parametric values
x=x(t)and y=y(t)associated with the curve Cand integrating with respect to the
parameter t. The above line integral is sometimes written in the form
∫
C
f(z)dz =
∫tb
ta
f(z(t)) z′(t)dt (12.187)
where z=z(t)is a parametric representation of the curve Cover the range ta≤t≤tb.
Whenever the curve C is not a smooth curve, but is composed of a finite num-
ber of arcs which are smooth, then the curve C is called piecewise smooth. If
C 1 , C 2 ,... , C mdenote the finite number of arcs over which the curve is smooth and
