Source: Homework Set 3, §13.1–13.4
Tags: arc length, arc length parametrisation, arc length function, speed, unit speed, Calculus III, MATH 215
Difficulty: Intermediate | Prerequisites: Integration techniques (especially trig identities and substitution), vector-valued functions (§13.1–13.2).
Arc length measures the actual distance along a curve, regardless of how fast a particle traverses it. This is the geometric length of the path. Arc length parametrisation is a way to reparametrise a curve so that the parameter itself measures distance along the curve, which simplifies many later formulas (curvature, for instance). If you are comfortable with the magnitude of a derivative and with definite integrals, you are ready.
The length of a parametric curve is the integral of |r'(t)| over the parameter interval. The arc length function s(t) accumulates distance from a starting point. If you can invert s(t) to get t(s), you can reparametrise the curve by arc length, giving a unit-speed traversal.
Arc length
The total distance along a curve between two points. Computed as the integral of the speed |r'(t)| over the parameter interval. Think of it as laying a string along the curve and measuring how long the string is.
Speed (of a parametrisation)
The magnitude |r'(t)| of the velocity vector. This tells you how fast a point is moving along the curve at parameter value t. Not the same as the velocity vector itself, which also carries directional information.
Arc length function s(t)
Defined as s(t) = ∫ from a to t of |r'(u)| du. It gives the distance travelled along the curve from the starting point r(a) to the current point r(t). In simple terms, s(t) is a running total of distance.
Arc length parametrisation
A reparametrisation of the curve where the parameter s represents distance along the curve. In an arc length parametrisation, |r'(s)| = 1 at every point, meaning the curve is traversed at unit speed. Think of it as a "constant pace" walk along the curve where each unit of parameter equals one unit of distance.
Reparametrisation
Replacing the parameter t with a new parameter (often s) via a substitution t = t(s). The curve itself does not change, only the "clock" that tracks position along it.
For a curve r(t) = ⟨f(t), g(t), h(t)⟩ on [a, b], the arc length is:
L = ∫ from a to b of |r'(t)| dt = ∫ from a to b of √(f'(t)² + g'(t)² + h'(t)²) dt.
The key step is computing |r'(t)| and simplifying before integrating. Trig identities are often essential.
r'(t) = ⟨3sin²(t)cos(t), −3cos²(t)sin(t), 2sin(t)cos(t)⟩.
|r'(t)|² = 9sin⁴(t)cos²(t) + 9cos⁴(t)sin²(t) + 4sin²(t)cos²(t).
Factor: = sin²(t)cos²(t)(9sin²(t) + 9cos²(t) + 4) = sin²(t)cos²(t)(9 + 4) = 13sin²(t)cos²(t).
So |r'(t)| = √13 · |sin(t)cos(t)| = (√13/2)|sin(2t)|.
On [0, π], sin(2t) changes sign at t = π/2. Split the integral:
L = (√13/2)[∫ from 0 to π/2 of sin(2t) dt + ∫ from π/2 to π of (−sin(2t)) dt] = (√13/2)[1 + 1] = √13.
Always simplify |r'(t)|² before taking the square root. Factor aggressively.
Watch for absolute values when simplifying square roots. |sin(t)cos(t)| ≠ sin(t)cos(t) everywhere.
Common identities that appear: sin²(t) + cos²(t) = 1, double angle sin(2t) = 2sin(t)cos(t), half-angle formulas.
Start with a parametric curve r(t) and compute the arc length function s(t) = ∫ from −∞ (or a) to t of |r'(u)| du.
Invert: solve for t as a function of s, giving t = t(s).
Substitute back: replace t with t(s) in the original parametrisation to get r(s) = r(t(s)).
The result is a curve parametrised by arc length, where |r'(s)| = 1.
Compute derivatives: x'(t) = −4t/(t² + 1)², y'(t) = 2(1 − t²)/(t² + 1)².
|r'(t)|² = [16t² + 4(1 − t²)²] / (t² + 1)⁴ = [16t² + 4 − 8t² + 4t⁴] / (t² + 1)⁴.
Simplify: = [4t⁴ + 8t² + 4] / (t² + 1)⁴ = 4(t² + 1)² / (t² + 1)⁴ = 4/(t² + 1)².
So |r'(t)| = 2/(t² + 1).
The arc length function: s(t) = ∫ from −∞ to t of 2/(u² + 1) du = 2 arctan(t) + π (since arctan(−∞) = −π/2).
Invert: arctan(t) = (s − π)/2, so t = tan((s − π)/2).
Substitute back into x(t) and y(t). Using the identity cos(θ) = (1 − tan²(θ/2))/(1 + tan²(θ/2)) and sin(θ) = 2tan(θ/2)/(1 + tan²(θ/2)) with appropriate substitution, you get x(s) = cos(s − π) = −cos(s) and y(s) = sin(s − π) = −sin(s), or equivalently x(s) = −cos(s), y(s) = −sin(s).
This is a unit circle. The original curve was a unit circle all along, just parametrised non-uniformly. The arc length parametrisation reveals it as uniform motion around the circle.
It separates geometry from speed. The shape of the curve is purely geometric; how fast you traverse it is a choice. Arc length parametrisation chooses unit speed.
Curvature formulas (§13.4) are simplest in arc length parametrisation: κ = |T'(s)|.
Not every curve has a closed-form arc length parametrisation. When it does, consider it a gift.
L = \int_a^b |\mathbf{r}'(t)|\, dt = \int_a^b \sqrt{f'(t)^2 + g'(t)^2 + h'(t)^2}\, dts(t) = \int_a^t |\mathbf{r}'(u)|\, du\frac{ds}{dt} = |\mathbf{r}'(t)| \quad \text{(speed)}In an arc length parametrisation: |r'(s)| = 1 for all s.
Arc length is how you measure the actual distance of a road, a wire, a pipe, or a flight path. GPS devices compute arc length along your route. In physics, if a particle moves along a curve, the arc length tells you the distance travelled regardless of how the motion was parametrised (fast, slow, or variable speed).
Students often forget the absolute value in |r'(t)| when simplifying. After taking a square root, you get |sin(t)cos(t)|, not sin(t)cos(t). Dropping the absolute value gives the wrong answer when the expression is negative.
Confusing the arc length (a number) with the arc length function (a function of t). The arc length of a curve on [a, b] is a single number L. The arc length function s(t) is a running total that depends on t.
Assuming that every curve can be reparametrised by arc length in closed form. Most cannot. The process is conceptually clean but the integrals are often intractable.
Treating the parameter t as distance. Unless you have specifically reparametrised by arc length, a change of 1 in t does not correspond to a distance of 1 along the curve.
⚠️ Arc length integrals are a staple exam problem. The main challenge is simplifying |r'(t)| before integrating. Practice trig identity manipulation.
⚠️ The full arc length parametrisation process (compute s(t), invert, substitute) is a multi-step problem that tests several skills at once. Know each step cold.
⚠️ Watch for absolute values. Splitting integrals at points where expressions change sign is frequently tested.
The arc length of a curve is computed by integrating ___ over the parameter interval.
True or false: In an arc length parametrisation, |r'(s)| can vary. ___
The arc length function s(t) represents ___.
True or false: To reparametrise by arc length, you must be able to invert s(t). ___
After simplifying |r'(t)|², you should take the square root and include ___.
Answers: 1. |r'(t)| (the speed). 2. False (|r'(s)| = 1 always). 3. The cumulative distance from the starting point to r(t). 4. True. 5. Absolute values where needed.
Q: Compute the arc length of r(t) = ⟨cos(t), sin(t), t⟩ on [0, 2π].
A: r'(t) = ⟨−sin(t), cos(t), 1⟩. |r'(t)| = √(sin²(t) + cos²(t) + 1) = √2. L = ∫ from 0 to 2π of √2 dt = 2π√2.
Q: For the curve x(t) = 2/(t² + 1) − 1, y(t) = 2t/(t² + 1), what is the speed |r'(t)|?
A: |r'(t)| = 2/(t² + 1). This comes from computing x'(t)² + y'(t)² and simplifying.
Q: After computing s(t) = 2arctan(t) + π, invert to find t as a function of s.
A: arctan(t) = (s − π)/2, so t = tan((s − π)/2).
Q: Why is arc length parametrisation useful for computing curvature?
A: In arc length parametrisation, curvature simplifies to κ = |T'(s)| = |r''(s)|. Without it, you need the more complex cross-product formula κ = |r'(t) × r''(t)| / |r'(t)|³.
Arc length connects directly to curvature (§13.4), which measures how much a curve bends per unit length. It also connects back to tangent vectors (§13.2), since the speed ds/dt = |r'(t)| is the magnitude of the tangent vector. In later chapters, arc length appears again in line integrals (Chapter 16), where you integrate a function along a curve weighted by arc length.
Arc length, arc length integral, arc length function, arc length parametrisation, reparametrisation, unit speed, speed of a curve, |r'(t)|, distance along a curve, curvature, trig identities, absolute value in integrals, MATH 215, Calculus III, §13.3