9.10. GEOMETRIC LENGTH OF A CURVE IN Rp 215

Proof: Using the theorems about the integral obtained earlier, in particular the funda-mental theorem of calculus,

ddx

(∫ x

0e−t2

dt)2

= 2(∫ x

0e−t2

dt)

e−x2= 2x

(∫ 1

0e−x2u2

du)

e−x2

= 2x∫ 1

0e−x2(u2+1)du

Then, integrating both sides and interchanging the order of integration with Fubini’s theo-rem, Theorem 9.9.4,(∫ x

0e−t2

dt)2

=∫ x

02v∫ 1

0e−v2(u2+1)dudv =

∫ 1

0

∫ x

02ve−v2(u2+1)dvdu

=∫ 1

0

−e−v2(u2+1)

u2 +1|x0du =

∫ 1

0

(1

u2 +1− e−x2(u2+1)

u2 +1

)du

Hence ∫ x

0e−t2

dt =

√√√√∫ 1

0

(1

u2 +1− e−x2(u2+1)

u2 +1

)du

Now the integrand on the right converges uniformly to 1u2+1 as x→ ∞ and so we can pass

to a limit as x→ ∞ and obtain limx→∞

∫ x0 e−t2

dt =√∫ 1

01

u2+1 du =√

π

2 .

9.10 Geometric Length of a Curve in Rp

I think that the right way to consider length is in terms of one dimensional Hausdorffmeasure. However, this is not a topic for this book. In this section, I am using the Euclideannorm because this is the one which corresponds to the usual notion of distance.

Definition 9.10.1 A set of points γ∗ ⊆ Rp is an oriented piecewise smooth curveif there is an oriented interval [a,b] and γ∗ ≡ γ ([a,b]) , and there are intermediate pointsbetween a and b,z1,z2, ...,zn such that (b−a)(zk− zk−1)> 0 and the following hold:

1. γ is one to one on [a,b) and γ is one to one on (a,b]

2. γ = γk on (zk−1,zk) with γ ′k continuous on [zk−1,zk] , where γ ′k (zk) ,γ′ (zk−1) is an

appropriate one sided derivative.

3. γ ′k ̸= 0 on (zk−1,zk) .

Here γ ′ is defined in the usual way, γ (t +h) = γ (t)+γ ′ (t)h+o(h). See Problem 16 onPage 153. Then letting P be an ordered partition of [a,b] ,P = {a = t0, t1, ..., tn = b} where(b−a)(tk− tk−1) > 0, and letting L(P) denote the sum ∑

nk=1 |γ (tk)− γ (tk−1)| , the length

of γ∗ denoted as L is defined as

sup{L(P) : P is an ordered partition of [a,b]} .

Note that this gives an intrinsic definition of length depending only on γ∗ and not onthe particular parametrization because it picks a particular order along the curve γ∗ andexpresses the length as the sup of the lengths of all polygonal approximations of this curve.