Difficulty: Intermediate Prerequisites: Calculus II arc length (Section 7.4 or equivalent), parametric equations, basic vector operations (dot product, magnitude).
Arc length measures the distance along a curve rather than the straight-line distance between two points. In Calculus II you learnt to compute arc length for flat curves, either as y = f(x) or as a 2D parametric path. Section 13.3 extends the same idea to three-dimensional space curves described by vector functions. If you are comfortable setting up and evaluating integrals involving derivatives, the jump to 3D is smaller than it looks: the formula gains one extra squared term under the radical, and the rest is the same machinery.
Arc length in 3D is the integral of the magnitude of the derivative of the position vector. The formula is a direct extension of the 2D parametric version, with a dz/dt term added under the square root. Most exam problems boil down to differentiating the components, plugging into the formula, and simplifying (often with a Pythagorean identity).
Arc length (s)
The total distance measured along a curve between two parameter values. In simple terms, if you walked along the curve with a pedometer, this is the number it would read.
Arc length integrand
The expression inside the arc length integral, equal to the magnitude of the velocity vector |r'(t)|. Think of it as the "speed" at which the tip of the position vector traces the curve.
Parametric arc length
The arc length of a curve described by parametric equations x(t), y(t) (and z(t) in 3D) rather than by y = f(x). This is the form you will use for all space curves in Calculus III.
Space curve
A curve that lives in three-dimensional space, described by a vector function r(t) =
Position vector r(t)
The vector-valued function that gives the coordinates of a point on the curve at parameter value t. Each component is a function of t.
Velocity vector r'(t)
The derivative of the position vector. Its magnitude gives the speed of a particle moving along the curve, and it is exactly the integrand in the arc length formula.
For a function y = f(x) between x = a and x = b, the arc length is the integral of sqrt(1 + (f'(x))^2) dx from a to b. This is the starting point; the parametric and 3D versions generalise it.
When a curve is given as x(t), y(t), the arc length over a <= t <= b is the integral of sqrt((dx/dt)^2 + (dy/dt)^2) dt. You differentiate each component with respect to the parameter, square each, add, take the square root, then integrate.
For a space curve r(t) =
s = integral from a to b of |r'(t)| dt
This expands to the integral of sqrt((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2) dt
The pattern extends to any dimension: add one squared derivative term per coordinate.
The key observation is that the integrand |r'(t)| equals the magnitude of the velocity vector. So computing arc length always follows the same steps: differentiate each component, compute the magnitude, integrate.
Differentiate: r'(t) = <2 cos(t), 5, -2 sin(t)>
Compute the magnitude squared: 4 cos^2(t) + 25 + 4 sin^2(t)
Factor: 25 + 4(sin^2(t) + cos^2(t)) = 25 + 4 = 29
The integrand simplifies to sqrt(29), a constant
Integrate: sqrt(29) times the length of the interval (20) = 20 sqrt(29)
The Pythagorean identity sin^2 + cos^2 = 1 did the heavy lifting. This is the single most common simplification trick in arc length problems involving trig parametrisations.
Differentiate: r'(t) = <1, 2t, 3t^2>
Magnitude squared: 1 + 4t^2 + 9t^4
The arc length integral is integral from 1 to 2 of sqrt(1 + 4t^2 + 9t^4) dt
This integral has no clean closed-form answer. On an exam, if you are given numerical bounds and the problem does not say "without a calculator", you may use one. Recognising that an integral is ugly and knowing when to stop is itself a skill the course tests.
s = \int_a^b \sqrt{1 + \bigl(f'(x)\bigr)^2}\, dx \quad \text{(single-variable, } y = f(x) \text{)}s = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\, dt \quad \text{(2D parametric)}s = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2}\, dt \quad \text{(3D parametric)}s = \int_a^b |\vec{r}\,'(t)|\, dt \quad \text{(compact vector form, works in any dimension)}The vector form is the one to memorise. Every other version is just the expanded form of |r'(t)| in however many dimensions you are working in.
Arc length is how engineers calculate the amount of cable needed to run along a curved path, or how much material is required to lay a pipeline over hilly terrain. In computer graphics, arc length parametrisation is used to move objects along curves at constant apparent speed.
Students often forget to differentiate the components before plugging into the formula. The integrand uses r'(t), not r(t).
A common error is squaring the entire magnitude expression instead of squaring each component separately and then summing. The square root goes around the sum of the squares, not around each term individually.
Students sometimes try to evaluate a messy arc length integral by hand when the problem allows a calculator. If the integrand does not simplify to something clean, that is often the signal that a numerical answer is acceptable.
Mixing up the bounds of integration with the endpoints of the curve in space. The bounds are parameter values (t = a to t = b), not coordinates.
The compact formula s = integral of |r'(t)| dt appears on virtually every Calculus III final.
Look for the Pythagorean identity: when the curve involves sin and cos, the integrand almost always simplifies. If it does not, double-check your derivatives.
"Set up but do not evaluate" problems test whether you can write the correct integral. Get the formula right even if the integral is ugly.
The relationship ds/dt = |r'(t)| is used again in the curvature derivation, so understanding it here pays off immediately.
True or false: The arc length integrand for a space curve is |r(t)|.
False. It is |r'(t)|, the magnitude of the derivative.
Fill in the blank: The compact vector formula for arc length is s = integral from a to b of _______ dt.
|r'(t)|
True or false: If sin^2(t) + cos^2(t) appears under the square root in an arc length integral, it simplifies to 1.
True.
Fill in the blank: The relationship ds/dt = _______ connects arc length to the parameter t.
|r'(t)|
True or false: You should always evaluate an arc length integral by hand on an exam.
False. Use a calculator for definite integrals unless the problem specifically says otherwise.
Q: Find the arc length of r(t) = <3t, 4t, 0> for 0 <= t <= 5.
A: r'(t) = <3, 4, 0>, so |r'(t)| = sqrt(9 + 16) = 5. The arc length is 5 times 5 = 25. (This is a straight line, so the result matches the distance formula.)
Q: Set up the arc length integral for r(t) =
A: r'(t) =
Q: Compute the arc length of r(t) =
A: r'(t) = <-sin(t), cos(t), 1>. |r'(t)| = sqrt(sin^2(t) + cos^2(t) + 1) = sqrt(2). Arc length = 2 pi sqrt(2).
Q: Why does the arc length formula use |r'(t)| rather than r'(t)?
A: r'(t) is a vector; you cannot integrate a vector to get a scalar distance. The magnitude |r'(t)| gives the scalar speed, and integrating speed over time gives the total distance travelled.
Q: A curve has r'(t) = <2 cos(t), 5, -2 sin(t)>. What is ds/dt?
A: ds/dt = |r'(t)| = sqrt(4 cos^2(t) + 25 + 4 sin^2(t)) = sqrt(29).
This connects directly to curvature (the next part of Section 13.3), because curvature is defined in terms of the arc length parameter s, and the derivative ds/dt = |r'(t)| derived here is the bridge between the two. Arc length also reappears in line integrals (Chapter 16), where you integrate a function along a curve with respect to arc length.
arc length, arc length formula, arc length integral, parametric arc length, space curve length, vector arc length, length of a curve, curve length calculus 3, MATH 163, Section 13.3, arc length integrand, ds/dt, magnitude of r prime, helix arc length, Pythagorean identity arc length, set up but do not evaluate