14.4. LINE INTEGRALS 267

In evaluating this replace the x in the formula for F with t, the y in the formula for Fwith cos(2t) and the z in the formula for F with sin(2t) because these are the values ofthese variables which correspond to the value of t. Taking the dot product, this equals thefollowing integral. ∫

π

0

(2t cos2t−2(sin2t) t2 +2cos2t

)dt = π

2

Example 14.4.12 Let C denote the oriented curve obtained by r (t) =(t,sin t, t3

)where

the orientation is determined by increasing t for t ∈ [0,2]. Also let F = (x,y,xz+ z). Find∫C F ·dR.

You use the definition.∫CF ·dR=

∫ 2

0

(t,sin(t) ,(t +1) t3) · (1,cos(t) ,3t2)dt

=∫ 2

0

(t + sin(t)cos(t)+3(t +1) t5

)dt =

125114− 1

2cos2 (2) .

Suppose you have a curve specified by r (s) = (x(s) ,y(s) ,z(s)) and it has the propertythat |r′ (s)| = 1 for all s ∈ [0,b]. Then the length of this curve for s between 0 and s1is∫ s1

0 |r′ (s)|ds =∫ s1

0 1ds = s1. This parameter is therefore called arc length because thelength of the curve up to s equals s. Now you can always change the parameter to be arclength.

Proposition 14.4.13 Suppose C is an oriented smooth curve parameterized by r (t) fort ∈ [a,b]. Then letting l denote the total length of C, there exists R(s), s ∈ [0, l] anotherparametrization for this curve which preserves the orientation and such that

∣∣R′ (s)∣∣ = 1so that s is arc length.

Prove: Let φ (t)≡∫ t

a |r′ (τ)|dτ ≡ s. Then s is an increasing function of t because

dsdt

= φ′ (t) =

∣∣r′ (t)∣∣> 0.

Now define R(s)≡ r(φ−1 (s)

). Then

R′ (s) = r′(φ−1 (s)

)(φ−1)′ (s) = r′

(φ−1 (s)

)∣∣r′ (φ−1 (s))∣∣

and so∣∣R′ (s)∣∣= 1 as claimed. R(l) = r

(φ−1 (l)

)= r

(φ−1(∫ b

a |r′ (τ)|dτ

))= r (b) and

R(0) = r(φ−1 (0)

)= r (a) and R delivers the same set of points in the same order as r

because dsdt > 0. ■

The arc length parameter is just like any other parameter, in so far as considerationsof line integrals are concerned, because it was shown above that line integrals are inde-pendent of parametrization. However, when things are defined in terms of the arc lengthparametrization, it is clear they depend only on geometric properties of the curve itself andfor this reason, the arc length parametrization is important in differential geometry.

Definition 14.4.14 As to piecewise smooth curves, recall these are just smooth curvesjoined together at a succession of points p1,p2, · · · ,pm. If C is such a curve which goesfrom p1 then to p2 then to p3 etc. one defines∫

CF ·d R≡

∫Cp1p2

F ·d R+∫

Cp2p3

F ·d R+ · · ·+∫

Cp(n−1)n

F ·dR