Difficulty: Intermediate | Prerequisites: Calculus II (Math 125) arc length in 2D, vector functions and derivatives (Ch. 13.1–13.2).
In Calculus II you learned to compute the arc length of a plane curve y = f(x). This chapter extends that idea to space curves described by vector functions r(t) = ⟨x(t), y(t), z(t)⟩. The arc length formula picks up a third component under the square root, and a new concept, curvature, lets you quantify how sharply a curve bends at any point. You should already be comfortable differentiating vector functions component by component and computing magnitudes of vectors in R³.
Arc length of a space curve is found by integrating the magnitude of the derivative of the position vector. You can reparametrise any curve so the parameter itself measures distance along the curve ("arc length parametrisation"), which simplifies later formulas for curvature and the unit normal vector.
Arc length
The total distance measured along a curve between two parameter values t = a and t = b. Think of it as laying a string along the curve and then measuring the string.
Arc length function s(t)
The function s(t) = ∫₀ᵗ |r'(u)| du that gives the arc length from the starting point (t = 0) to a general parameter value t. In simple terms, this is a running total of how far you have travelled along the curve.
Reparametrisation by arc length
Rewriting a vector function r(t) so the parameter is arc length s rather than an arbitrary t. The result is a curve whose speed |r'(s)| = 1 at every point. Think of it as switching from "how much time has passed" to "how far have I walked."
Speed
The magnitude |r'(t)| of the velocity vector. Speed tells you how fast the point moves along the curve for a given change in t.
In Calculus II (Math 125), arc length for a plane curve with parametric equations x = x(t), y = y(t) was given by the integral of √[(x'(t))² + (y'(t))²] from a to b.
In three dimensions, the curve r(t) = ⟨x(t), y(t), z(t)⟩ adds a z-component. The arc length from t = a to t = b is:
s = ∫ₐᵇ √[(x'(t))² + (y'(t))² + (z'(t))²] dt = ∫ₐᵇ |r'(t)| dt
The integrand |r'(t)| is the speed of the particle tracing out the curve.
Let r(t) = ⟨4 sin t, 3t + 7, 4 cos t⟩. Find the arc length from t = 1 to t = 8.
Compute the derivative: r'(t) = ⟨4 cos t, 3, −4 sin t⟩.
Find the magnitude: |r'(t)| = √[16 cos²t + 9 + 16 sin²t] = √[16(cos²t + sin²t) + 9] = √[16 + 9] = √25 = 5.
Integrate: s = ∫₁⁸ 5 dt = 5t |₁⁸ = 5(8) − 5(1) = 35.
The constant speed (5) makes this integral straightforward. Many exam problems are designed so that the speed simplifies neatly.
Goal: rewrite r(t) as r(t(s)) so the new parameter s measures distance along the curve and the speed is always 1.
Process:
Compute the arc length function: s(t) = ∫₀ᵗ |r'(u)| du.
Solve for t in terms of s.
Substitute back into r(t) to get r(s).
Using the same r(t) = ⟨4 sin t, 3t + 7, 4 cos t⟩ from above.
Step 1: s(t) = ∫₀ᵗ 5 du = 5t.
Step 2: t = s/5.
Step 3: r(s) = ⟨4 sin(s/5), 3s/5 + 7, 4 cos(s/5)⟩.
Verify: |r'(s)| should equal 1. Taking the derivative with respect to s gives r'(s) = ⟨(4/5) cos(s/5), 3/5, −(4/5) sin(s/5)⟩, and |r'(s)| = √[16/25 + 9/25] = √[25/25] = 1. Confirmed.
s = \int_a^b \sqrt{[x'(t)]^2 + [y'(t)]^2 + [z'(t)]^2}\, dt = \int_a^b |\mathbf{r}'(t)|\, dts(t) = \int_0^t |\mathbf{r}'(u)|\, duTo reparametrise: solve s(t) for t, then substitute into r(t).
Arc length calculations are used in engineering to determine the length of wire, cable, or piping that follows a curved path through three-dimensional space. Reparametrisation by arc length is the foundation of CNC machining: the cutting tool must move at constant speed along a curved surface, so the machine's computer converts the toolpath to an arc-length parameter.
Students often forget to take the derivative of each component before plugging into the arc length formula. The formula uses r'(t), not r(t).
When the speed |r'(t)| simplifies to a constant (as in the helix example), students sometimes doubt their algebra. A constant speed is a strong hint that the problem is set up correctly, not a sign of error.
Reparametrisation does not change the shape of the curve. The curve itself is identical; only the way you label points along it changes.
Students sometimes try to reparametrise by just replacing every t with s. You must first find the relationship s(t) and invert it.
⚠️ The arc length integral appears on nearly every Calc III exam. Know the formula cold and practise simplifying the expression under the square root using trig identities.
⚠️ Reparametrisation problems typically give you a curve whose speed is constant, so the integral is easy to evaluate. If you get a messy integral, re-check your derivative.
⚠️ You will need the arc length function s(t) again when computing curvature, so make sure you understand the process before moving on.
True or False: The arc length formula in R³ is identical to the R² formula except it includes a (z'(t))² term under the square root. (True)
Fill in the blank: If |r'(t)| = 7 for all t, the arc length from t = 2 to t = 5 is ___. (21)
True or False: Reparametrising a curve by arc length changes the shape of the curve. (False)
Fill in the blank: After reparametrisation by arc length, the speed |r'(s)| always equals ___. (1)
True or False: The arc length function s(t) is always non-decreasing. (True, because |r'(t)| ≥ 0)
Q: Find the arc length of r(t) = ⟨2t, t², t³⟩ from t = 0 to t = 1.
A: Compute r'(t) = ⟨2, 2t, 3t²⟩. Then |r'(t)| = √(4 + 4t² + 9t⁴). The arc length is ∫₀¹ √(4 + 4t² + 9t⁴) dt. (This integral does not simplify to a closed form and would likely be left as a set-up-only problem or evaluated numerically.)
Q: Let r(t) = ⟨cos t, sin t, t⟩. Reparametrise with respect to arc length starting from t = 0.
A: r'(t) = ⟨−sin t, cos t, 1⟩, so |r'(t)| = √(sin²t + cos²t + 1) = √2. Then s = √2 · t, giving t = s/√2. The reparametrised curve is r(s) = ⟨cos(s/√2), sin(s/√2), s/√2⟩.
Q: A curve has |r'(t)| = 3 for all t. What is the arc length from t = 4 to t = 10?
A: s = ∫₄¹⁰ 3 dt = 3(10 − 4) = 18.
Q: Why does reparametrisation by arc length guarantee unit speed?
A: Because ds/dt = |r'(t)|, the chain rule gives |dr/ds| = |r'(t)| / (ds/dt) = |r'(t)| / |r'(t)| = 1.
This material connects directly to curvature (Ch. 13.3 continued), which is defined in terms of the arc length parameter. The unit tangent vector T(t) = r'(t)/|r'(t)| from Ch. 13.2 reappears here because curvature measures how fast T changes with respect to arc length. Later in Ch. 13.4, the TNB frame (tangent, normal, binormal) builds on arc length parametrisation to describe the geometry of a curve in full.
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