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There are examples of paths whose supports, for example, containan entire three-dimensional cube (the so-called Peano “curves”). However, if thefunctions x(t), y(t), and z(t) are sufficiently regular (as happens, for example, inthe case of a mechanical motion, when they are differentiable), we are guaranteedthat nothing contrary to our intuition will occur, as one can verify rigorously.Definition 5 A path Γ : I → R3 for which the mapping I → Γ (I ) is one-to-one iscalled a simple path or parametrized curve, and its support is called a curve in R3 .Definition 6 A closed path Γ : [a, b] → R3 is called a simple closed path or simpleclosed curve if the path Γ : [a, b[ → R3 is simple.Thus a simple path differs from an arbitrary path in that when moving over itssupport we do not return to points reached earlier, that is, we do not intersect ourtrajectory anywhere except possibly at the terminal point, when the simple path isclosed.Definition 7 The path Γ : I → R3 is called a path of a given smoothness if thefunctions x(t), y(t), and z(t) have that smoothness.(For example, the smoothness C[a, b], C (1) [a, b], or C (k) [a, b].)Definition 8 A path Γ : [a, b] → R3 is piecewise smooth if the closed interval [a, b]can be partitioned into a finite number of closed intervals on each of which thecorresponding restriction of the mapping Γ is defined by continuously differentiablefunctions.It is smooth paths, that is, paths of class C (1) and piecewise smooth paths that weintend to study just now.Let us return to the original problem, which we can now state as the problem ofdefining the length of a smooth path Γ : [a, b] → R3 .Our initial ideas about the length l[a, b] of the path traversed during the timeinterval α ≤ t ≤ β are as follows: First, if α < β < γ , thenl[α, γ ] = l[α, β] + l[β, γ ],and second, if v(t) = (ẋ(t), ẏ(t), ż(t)) is the velocity of the point at time t, theninf v(t)(β − α) ≤ l[α, β] ≤ sup v(t)(β − α).x∈[α,β]x∈[α,β]Thus, if the functions x(t), y(t), and z(t) are continuously differentiable on[a, b], by Proposition 1 we arrive in a deterministic manner at the formula b bl[a, b] =v(t) dt =ẋ 2 (t) + ẏ 2 (t) + ż2 (t) dt,(6.51)aa6.4 Some Applications of Integration379which we now take as the definition of the length of a smooth path Γ : [a, b] → R3 .If z(t) ≡ 0, the support lies in a plane, and formula (6.51) assumes the formbl[a, b] =ẋ 2 (t) + ẏ 2 (t) dt.(6.52)aExample 3 Let us test formula (6.52) on a familiar object.

Suppose the point movesaccording to the lawx = R cos 2πt,y = R sin 2πt.(6.53)Over the time interval [0, 1] the point will traverse a circle of radius R, that is, apath of length 2πR if the length of a circle can be computed from this formula.Let us carry out the computation according to formula (6.52):l[0, 1] =1#(−2πR sin 2πt)2 + (2πR cos 2πt)2 dt = 2πR.0Despite the encouraging agreement of the results, the reasoning just carried outcontains some logical gaps that are worth paying attention to.The functions cos α and sin α, if we use the high-school definition of them, arethe Cartesian coordinates of the image p of the point p0 = (1, 0) under a rotationthrough angle α.Up to sign, the quantity α is measured by the length of the arc of the circlex 2 + y 2 = 1 between p0 and p.

Thus, in this approach to trigonometric functionstheir definition relies on the concept of the length of an arc of a circle and hence, incomputing the circumference of a circle above, we were in a certain sense completing a logical circle by giving the parametrization in the form (6.53).However, this difficulty, as we shall now see, is not fundamental, since aparametrization of the circle can be given without resorting to trigonometric functions at all.Let us consider the problem of computing the length of the graph of a functiony = f (x) defined on a closed interval [a, b] ⊂ R.

We have in mind the computationof the length of the path Γ : [a, b] → R2 having the special parametrizationx → x, f (x) ,from which one can conclude that the mapping Γ : [a, b] → R2 is one-to-one.Hence, by Definition 5 the graph of a function is a curve in R2 .In this case formula (6.52) can be simplified, since by setting x = t and y = f (t)in it, we obtain b)*2l[a, b] =1 + f (t) dt.(6.54)a3806IntegrationIn particular, if we consider the semicircle#y = 1 − x 2 , −1 ≤ x ≤ 1,of the circle x 2 + y 2 = 1, we obtain for it'$%2 1 +1−xdx1+ √dx =.l=√21−x1 − x2−1−1(6.55)But the integrand in this last integral is an unbounded function, and hence doesnot exist in the traditional sense we have studied.

Does this mean that a semicirclehas no length? For the time being it means only that this parametrization of thesemicircle does not satisfy the condition that the functions ẋ and ẏ be continuous,under which formula (6.52), and hence also formula (6.54), was written. For thatreason we must either consider broadening the concept of integral or passing to aparametrization satisfying the conditions under which (6.54) can be applied.We remark that if we consider this parametrization on any closed interval of theform [−1 + δ, 1 − δ], where −1 < −1 + δ < 1 − δ < 1, then formula (6.54) applieson that interval, and we find the length 1−δdxl[−1 + δ, 1 − δ] =√1 − x2−1+δfor the arc of the circle lying above the closed interval [−1 + δ, 1 − δ].It is therefore natural to consider that the length l of the semicircle is the limitlimδ→+0 l[−1 + δ, 1 − δ].

One can interpret the integral in (6.55) in the same sense.We shall study this naturally arising extension of the concept of a Riemann integralin the next section.As for the particular problem we are studying, without even changing theparametrization one can find, for example, the length l[− 12 , 12 ] of an arc of the unitcircle subtended by a chord congruent to the radius of the circle. Then (from geometric considerations alone) it must be that l = 3 · l[− 12 , 12 ].We remark also that #(1 − x 2 + x 2 ) dx1dxx d(1 − x 2 )==1 − x 2 dx −=√√√21 − x21 − x21 − x2 ##1 − x 2 dx − x 1 − x 2 ,=2and thereforel[−1 + δ, 1 − δ] = 21−δ−1+δ#1−δ #1 − x 2 dx − x 1 − x 2 −1+δ .Thus,l = lim l[−1 + δ, 1 − δ] = 2δ→+01−1#1 − x 2 dx.6.4 Some Applications of Integration381The length of a semicircle of unit radius is denoted π , and we thus arrive at thefollowing formula 1#π =21 − x 2 dx.−1This last integral is an ordinary (not generalized) Riemann integral and can becomputed with any precision.If for x ∈ [−1, 1] we define arccos x as l[x, 1], then by the computations carriedout above 1dtarccos x =,√1 − t2xor 1##arccos x = x 1 − x 2 + 21 − t 2 dt.xIf we regard arc length as a primitive concept, then we must also regard thefunction x → arccos x introduced just now and the function x → arcsin x, which canbe introduced similarly, as primitive.

But the functions x → cos x and x → sin x canthen be obtained as the inverses of these functions on the corresponding intervals.In essence, this is what is done in elementary geometry.The example of the length of a semicircle is instructive not only because whilestudying it we made a remark on the definition of the trigonometric functions thatmay be of use to someone, but also because it naturally raises the question whetherthe number defined by formula (6.51) depends on the coordinate system x, y, z andthe parametrization of the curve when one is finding the length of a curve.Leaving to the reader the analysis of the role played by three-dimensional Cartesian coordinates, we shall examine here the role of the parametrization.We need to clarify that by a parametrization of a curve in R3 , we mean a definition of a simple path Γ : I → R3 whose support is that curve.

The point or numbert ∈ I is called a parameter and the interval I the domain of the parameter.If Γ : I → L and Γ. : I˜ → L are two one-to-one mappings with the same set ofvalues L, there naturally arise one-to-one mappings Γ˜ −1 ◦ Γ : I → I˜ and Γ −1 ◦ Γ. :I˜ → I between the domains I and I˜ of these mappings.In particular, if there are two parametrizations of the same curve, then there isa natural correspondence between the parameters t ∈ I and τ ∈ I˜, t = t (τ ) or τ =τ (t), making it possible, knowing the parameter of a point in one parametrization,to find its parameter in the other parametrization.Let Γ : [a, b] → L and Γ. : [α, β] → L be two parametrizations of the samecurve with the correspondences Γ (a) = Γ.(α) and Γ (b) = Γ.(β) between their initial and terminal points.

Then the transition functions t = t (τ ) and τ = τ (t) fromone parameter to another will be continuous, strictly monotonic mappings of theclosed intervals a ≤ t ≤ b and α ≤ τ ≤ β onto each other with the initial points andterminal points corresponding: a ↔ α, b ↔ β.Here, if the curves Γ and Γ. are defined by the triples (x(t), y(t), z(t)) and(x̃(t), ỹ(t), z̃(t)) of smooth functions such that |v(t)|2 = ẋ 2 (t) + ẏ 2 (t) + ż2 (t) = 03826Integrationon [a, b] and ṽ(τ )|2 = x̃˙ 2 (τ ) + ỹ˙ 2 (τ ) + z̃˙ 2 (τ ) = 0 on [α, β], then one can verifythat in this case the transition functions t = t (τ ) and τ = τ (t) are smooth functionshaving positive derivatives on the intervals on which they are defined.We shall not undertake to verify this assertion here; it will eventually be obtainedas a corollary of the important implicit function theorem.

At the moment this assertion is mostly intended as motivation for the following definition.Definition 9 The path Γ. : [α, β] → R3 is obtained from Γ : [a, b] → R3 by anadmissible change of parameter if there exists a smooth mapping T : [α, β] → [a, b]such that T (α) = a, T (β) = b, T (τ ) > 0 on [α, β] and Γ. = Γ ◦ T .We now prove a general proposition.Proposition 2 If a smooth path Γ. : [α, β] → R3 is obtained from a smooth pathΓ : [a, b] → R3 by an admissible change of parameter, then the lengths of the twopaths are equal.Proof Let Γ. : [α, β] → R3 and Γ : [a, b] → R3 be defined respectively by thetriples of smooth functions τ → (x̃(τ ), ỹ(τ ), z̃(τ )) and t → (x(t), y(t), z(t)), andlet t = t (τ ) be the admissible change of parameter under whichx̃(τ ) = x t (τ ) ,ỹ(τ ) = y t (τ ) ,z̃(τ ) = z t (τ ) .Using the definition (6.51) of path length, the rule for differentiating a compositefunction, and the rule for change of variable in an integral, we have bẋ 2 (t) + ẏ 2 (t) + ż2 (t) dt =a= ẋ 2 t (τ ) + ẏ 2 t (τ ) + ż2 t (τ ) t (τ ) dτ =ββα=βα=α) *2 ) *2 ) *2ẋ t (τ ) t (τ ) + ẏ t (τ ) t (τ ) + ż t (τ ) t (τ ) dτ =x̃˙ 2 (τ ) + ỹ˙ 2 (τ ) + z̃˙ 2 (τ ) dτ.Thus, in particular, we have shown that the length of a curve is independent of asmooth parametrization of it.The length of a piecewise smooth path is defined as the sum of the lengths of thesmooth paths into which it can be divided; for that reason it is easy to verify that thelength of a piecewise smooth path also does not change under an admissible changeof its parameter.To conclude the discussion of the concept of the length of a path and the lengthof a curve (which, after Proposition 2, we now have the right to talk about), weconsider another example.6.4 Some Applications of Integration383Fig.

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