Mathematics
Arc Length of a Curve
The arc length of a curve is the distance along the curve between two points. It is calculated by integrating the square root of the sum of the squares of the derivatives of the curve with respect to the variable of integration. The arc length formula is used in calculus to find the length of a curve.
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7 Key excerpts on "Arc Length of a Curve"
- James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
But that might be dif- ficult to do with much accuracy if we have a complicated curve. We need a precise defini- tion for the length of an arc of a curve, in the same spirit as the definitions we developed for the concepts of area and volume. ■ Arc Length of a Curve If a curve is a polygon, we can easily find its length; we just add the lengths of the line segments that form the polygon. (We can use the distance formula to find the distance between the endpoints of each segment.) We are going to define the length of a general curve by first approximating it by a polygonal path (a path consisting of connected line segments) and then taking a limit as the number of segments of the path is increased. This process is familiar for the case of a circle, where the circumference is the limit of lengths of inscribed polygons (see Figure 2). Now suppose that a curve C is defined by the equation y - f s xd, where f is continu- ous and a < x < b. We obtain a polygonal approximation to C by dividing the interval fa, bg into n subintervals with endpoints x 0 , x 1 , . . . , x n and equal width Dx. If y i - f s x i d, then the point P i s x i , y i d lies on C and the polygonal path with vertices P 0 , P 1 , . . . , P n , illustrated in Figure 3, is an approximation to C. y P¸ P¡ P™ P i-1 P i P n y=ƒ 0 x i ¤ i-1 b x¡ a x x The length L of C is approximately the length of this polygonal path and the approxi- mation gets better as we let n increase. (See Figure 4, where the arc of the curve between P i21 and P i has been magnified and approximations with successively smaller values of Dx are shown.) Therefore we define the length L of the curve C with equation y - f s xd, a < x < b, as the limit of the lengths of these approximating polygonal paths (if the limit exists): 1 L - lim n l ` o n i-1 | P i21 P i | where | P i21 P i | is the distance between the points P i21 and P i .
eBook - PDF- Tom M. Apostol(Author)
- 2019(Publication Date)
- Wiley(Publisher)
(b) Prove that the corresponding principal normal vectors N X and N Y satisfy N Y (t) = N X [u(t)] at each point of C. Deduce that the osculating plane is invariant under a change of parameter. 14.10 The definition of arc length Various parts of calculus and analytic geometry refer to the Arc Length of a Curve. Before we can study the properties of the length of a curve we must agree on a definition of arc length. The purpose of this section is to formulate such a definition. This will lead, in a natural way, to the construction of a function (called the arc-length function) which measures the length of the path traced out by a moving particle at every instant of its motion. Some of the basic properties of this function are discussed in Section 14.12. In particular, we shall prove that for most curves that arise in practice this function may be expressed as the integral of the speed. To arrive at a definition of what we mean by the length of a curve, we proceed as though we had to measure this length with a straight yardstick. First, we mark off a number of points on the curve which we use as vertices of an inscribed polygon. (An example is shown in Figure 14.12.) Then, we measure the total length of this polygon with our yardstick and consider this as an approximation to the length of the curve. We soon observe that some polygons “fit” the curve better than others. In particular, if we start with a polygon P 1 , and construct a new inscribed polygon P 2 by adding more vertices to those of P 1 , it is clear that the length of P 2 will be larger than that of P 1 , as suggested in Figure 14.13. In the same way we can form more and more polygons with successively larger and larger lengths. On the other hand, our intuition tells us that the length of any inscribed polygon should not exceed that of the curve (since a straight line is the shortest path between two points).
eBook - PDFCalculus
Multivariable
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Theorem 12.3.1 If C is the graph in 2-space or 3-space of a smooth vector-valued function r(t), then its arc length L from t = a to t = b is L = b a dr dt dt (5) Example 2 Find the arc length of that portion of the circular helix x = cos t, y = sin t, z = t from t = 0 to t = π. Solution Set r(t) = (cos t)i + (sin t)j + tk = cos t, sin t, t. Then r (t) = −sin t, cos t, 1 and r (t) = (− sin t) 2 + (cos t) 2 + 1 = √ 2 From Theorem 12.3.1 the arc length of the helix is L = π 0 dr dt dt = π 0 √ 2 dt = √ 2π Arc Length as a Parameter For many purposes the best parameter to use for representing a curve in 2-space or 3-space para- metrically is the length of arc measured along the curve from some fixed reference point. This can be done as follows: Using Arc Length as a Parameter Step 1. Select an arbitrary point on the curve C to serve as a reference point. Step 2. Starting from the reference point, choose one direction along the curve to be the positive direction and the other to be the negative direction. Step 3. If P is a point on the curve, let s be the “signed” arc length along C from the reference point to P, where s is positive if P is in the positive direction from the reference point and s is negative if P is in the negative direction. Figure 12.3.2 illustrates this idea. − d ir e c tion Reference point C s = −3 s = −2 s = −1 s = 1 s = 2 s = 3 + d i r e c t i o n FIGURE 12.3.2 By this procedure, a unique point P on the curve is determined when a value for s is given. For example, s = 2 determines the point that is 2 units along the curve in the positive direction from the reference point, and s = − 3 2 determines the point that is 3 2 units along the curve in the negative direction from the reference point. Let us now treat s as a variable. As the value of s changes, the corresponding point P moves along C and the coordinates of P become functions of s. Thus, in 2-space the coordinates of P are
eBook - PDFCalculus
Late Transcendentals
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
This can be done as follows: Using Arc Length as a Parameter Step 1. Select an arbitrary point on the curve C to serve as a reference point. Step 2. Starting from the reference point, choose one direction along the curve to be the positive direction and the other to be the negative direction. Step 3. If P is a point on the curve, let s be the “signed” arc length along C from the reference point to P, where s is positive if P is in the positive direction from the reference point and s is negative if P is in the negative direction. Figure 12.3.2 illustrates this idea. Figure 12.3.2 By this procedure, a unique point P on the curve is determined when a value for s is given. For example, s = 2 determines the point that is 2 units along the curve in the positive direction from the reference point, and s = − 3 2 determines the point that is 3 2 units along the curve in the negative direction from the reference point. Let us now treat s as a variable. As the value of s changes, the corresponding point P moves along C and the coordinates of P become functions of s. Thus, in 2-space the coordinates of P are (x(s), y(s)), and in 3-space they are (x(s), y(s), z(s)). Therefore, in 2-space or 3-space the curve C is given by the parametric equations x = x (s), y = y (s) or x = x (s), y = y (s), z = z (s) A parametric representation of a curve with arc length as the parameter is called an arc length parametrization of the curve. Note that a given curve will generally have infinitely many different arc length parametrizations, since the reference point and orientation can be chosen arbitrarily. Example 3 Find the arc length parametrization of the circle x 2 + y 2 = a 2 with counterclockwise orientation and (a, 0) as the reference point. Solution. The circle with counterclockwise orientation can be represented by the para- metric equations x = a cos t, y = a sin t (0 ≤ t ≤ 2π ) (6)
eBook - PDF- Brian H. Chirgwin, Charles Plumpton(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
Ί Examples, (i) Find the length of the curve y = —log (1 — x 2 ) from the origin to the point where x = x 1 (0 < x x < 1). dy/άχ = 2a;/(l — x 2 ) > 0 for 0 < x < 1. x x x t ,. arc length -J | / [ l + j ^ f ] d, -/ / { j ^ g } d. 0 0 -Jr=i d * -/(-1 + ïhï) d * = log (r^f ) -*■ o o (ii) Calculate the length from x = 0 to x = απ oi the curve whose parametric equations are x = α(Θ — sin Θ), y = α(1 — cos 0). # = α(1 — cos Θ) > 0 for 0 < 0 < π. n .·. arc length = ^]/{α 2 (1 - cos Θ) 2 + a 2 sin 2 0} do o n π = aj]/{2(l -coso)} d0 = 2afsm(±6)de = 4a. o o Exercises 4:4 1. Show that the length of the arc of the parabola x = at 2 , y = 2at between the limits t = — 1 and t = 1 is 2a[}/2 + log(l +l/2)]. 120 A COÏÏBSE OF MATHEMATICS 2. Find the area of the region bounded by two arcs of the parabolas y 2 = x, x 2 = y, and show that the perimeter of this region is ilog(2 + )/5) + )/5. 3. The coordinates of a point on a curve are given by the equations x = a(u — tanh u), y = a sech u where u is a variable parameter. Prove that the arc s of the curve, measured from a suitable point (to be specified), is given by s = a log cosh u . 4. Prove that the curve x = a cos 3 £, y = 2a sin 3 £ has length 28 aß. 5. Prove the length of the arc of the curve y = (* + l)(* + 2 ) -I log (2x + 3) between the points for which x = 1 and x = 2, respectively, is 6 + | l o g e | . 6. Sketch the curve whose parametric equations are x = t 2 , y = (t— l) 2 . Calculate (i) the area bounded by this curve and the chord x = 1, (ii) the length of the curve between the points where t = 0 and t — 1. 4:5 Curvature The direction of a curve at any point is determined by the gradient dy/dx = tan w. The curvature, which we denote by κ, measures the rate of change of direction of the curve w. r. to arc length and is defined κ = ^ 7 · < 4 · 8 > If we consider the circle of radius a with its centre at (0, a), see Fig. 18, the arc length OP is αψ, and the intrinsic equation of the circle is s = αψ.
eBook - PDFMathematical Methods
A Course of Mathematics for Engineers and Scientists
- Brian H. Chirgwin, Charles Plumpton(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
§ 4 : 4] GEOMETRY OF TWO DIMENSIONS 185 Therefore the required arc length 0 0 -/-!ÎF*-jV-râr)«--(-&H· 0 0 Example 2. Calculate the length from x = 0 to x = απ of the curve, the cycloid, whose parametric equations are x = a(0-sin Θ), y = «(1-cos 0). x = 0(1 -cos Θ) > 0 for 0 < θ < ,τ. π Therefore the required arc length = J Z{tf 2 0 -cos θ) 2 + α 2 sin 2 0} d# o n n = a j V{2(1 -cos Ô)} d0 = 2a j sin (10) dö = 4a. 0 0 Exercises 4:4 1. Show that the length of the arc of the parabola x = at 2 , y = 2at be-tween the limits / = -1 and / = 1 is 2e[V2 + ln(l + V2)]. 2. Find the area of the region bounded by two arcs of the parabolas y 2 = x, x 2 = y, and show that the perimeter of this region is j l n ( 2 + V 5 ) + V 5 . 3. The coordinates of a point on a curve are given by the equations x = a(u — tanh u), y = a sech u where u is a variable parameter. Prove that the arc s of the curve, measured from a suitable point (to be specified), is given by s = a In cosh u. 4. Prove that the curve x = a cos 3 /, y = 2a sin 3 1 has length 28#/3. 5. Prove that the length of the arc of the curve y = (jc+l)(x + 2)-iln(2*+3) between the points for which x = 1 and x = 2, respectively, is 6 + i l n J . 6. Sketch the curve whose parametric equations are x = t y = (t-l) 2 . (Continued overleaf)
eBook - PDFApplied Calculus
The Commonwealth and International Library: Programmed Texts Series
- D. J. Bell, F. H. George(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
The radius of curvature is usually denoted by p. The curva-ture is then given by 1/p. Since the curvature is given by άφ/ds we have ds If the curve AB is a circle of radius r, what is the curvature of AB at a point on it in terms of r? 183 -. r 187 r. 175 from 172 There are several formulae for the radius of curvature. We shall look first at the formula ds Ρ = άφ which we have met already. This formula is used when the curve is given by an equation involving the variables 5 and φ. Such an equation is called the intrinsic equation of the curve. What is the curvature at a point on the catenary s = k. tan ψΊ 188 cos 2 ψ/k. 191 k . sec 2 φ. 194 k . log(sec ψ). 176 from 168 Here are the answers to the examples set : (a) (b) (c) (d) (-1,-2) (0,0) (1,2) (-2, 12) (1,3) (3, 27) (1,0) (e,D (0,0) (π/2,1) (π/4,1/V2) tangent 6x — y — —4 y = 0 6x — y = 4 12x + y = -12 6x — y = 3 18x -y = 27 x — y = 1 x — ey = 0 )> = X x -V 2 y = ^ -l normal x + ey = -13 JC = 0 x + 6y = 13 x - 12)/ = -146 x + 6y = 19 x + 18y = 489 x + )> = 1 ex + y = 1 + e 2 x + 3; = 0 x = π/2 2x + 72y = 1 + ^ When you are satisfied that you have obtained the correct answers, we can proceed to the next topic which is the length of arcs of curves. Turn back to 158 177 from 192 To solve the integral sinh-U cosh 2 u du, o we use the identity cosh 2u = 2 cosh 2 u — 1. The integrand of the above integral then becomes ^(cosh 2M + 1) and the integral sinh _ 1 1 (cosh 2u + 1) du. 0 Straightforward integration yields sinh »1 £[! sinh 2w + u] o sinh U = |[sinh u . cosh u + u] o = i(V2 + sinh 4} and this is the length of the curve y = x 2 from x = 0 to x = . Now we will see if you can do a similar example yourself. Turn to 181 178 from 174 Your answer was i{yj2 + logi^ + 1)}. Very good indeed. You are perfectly correct and you did well to obtain the right answer. Thus the length of the curve y = x 2 from x = 0 to x = is ÜJ2 + log (y/2+n). Now we will see if you can do a similar example yourself. Turn to 181 179
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