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Arc Lengths of Curves
Parametric Equations

Learn to compute the exact length of a curve defined by parametric equations using integration and the arc length formula.

AP Calculus BC

What Are Parametric Curves?

Instead of writing \( y = f(x) \), we can describe a curve by letting both coordinates depend on a third variable called a parameter, usually \( t \):

This representation is powerful because it can describe curves that loop back, self-intersect, or trace paths that fail the vertical line test.

Why Measure Arc Length?

Physics
Distance traveled by a particle moving along a curved trajectory
Engineering
Length of wire, road, or pipe following a curved design path
Geometry
Circumference of ellipses, spirals, and other curves without simple formulas

The Infinitesimal Approach

\(dx\) \(dy\) \(ds\) Tiny segment approximated by a straight line

Deriving the Arc Length Formula

1
Start with the distance element: \( ds = \sqrt{dx^2 + dy^2} \)
2
Factor out \( dt \): \( ds = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \; dt \)
3
Integrate over the parameter interval: \( L = \displaystyle\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \; dt \)

The Core Formula

\(t = a\) \(t = b\) Arc Length \(L\) x y

The green arc between \( t = a \) and \( t = b \) has length \( L \).

When Does the Formula Apply?

1
Continuity: \( f'(t) \) and \( g'(t) \) must be continuous on \( [a, b] \).
2
No retracing: The curve should be traversed exactly once as \( t \) goes from \( a \) to \( b \). (If the curve doubles back, you may count segments more than once.)
3
Not both zero simultaneously: \( f'(t) \) and \( g'(t) \) should not both equal zero at the same \( t \) (no cusps).

Parametric vs. Rectangular Arc Length

Parametric Rectangular \( y = f(x) \)
Formula \( \displaystyle\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\, dt \) \( \displaystyle\int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2}\, dx \)
Variable Integrate with respect to \( t \) Integrate with respect to \( x \)
Handles loops? Yes No (fails vertical line test)
Connection Set \( x = t \), \( y = f(t) \): then \( dx/dt = 1 \) recovers the rectangular formula

How to Apply the Formula

1
Identify \( x(t) \) and \( y(t) \): Write down the parametric equations and the interval \( [a, b] \).
2
Differentiate: Compute \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \).
3
Square and add: Form \( \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 \).
4
Simplify under the radical: Factor, use identities, or combine terms.
5
Integrate: Evaluate \( \displaystyle\int_a^b \sqrt{\text{expression}} \; dt \). Use substitution or known integrals as needed.

Example 1: Line Segment

Find the arc length of \( x = 1 + 3t \), \( y = 2 + 4t \) for \( 0 \leq t \leq 1 \).

1
\( \frac{dx}{dt} = 3 \), \( \frac{dy}{dt} = 4 \)
2
\( \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 = 9 + 16 = 25 \)
3
\( L = \displaystyle\int_0^1 \sqrt{25}\, dt = \int_0^1 5\, dt = 5 \)

Example 2: Circle of Radius \( r \)

Verify the circumference formula using \( x = r\cos t \), \( y = r\sin t \) for \( 0 \leq t \leq 2\pi \).

\(r\) \(t=0\)

Example 3: Cycloid

Find the arc length of one arch of the cycloid: \( x = t - \sin t \), \( y = 1 - \cos t \), for \( 0 \leq t \leq 2\pi \).

peak \(t = 0\) \(t = 2\pi\)

Key Identity: Half-Angle Trick

In the cycloid example, simplifying the integrand required a crucial identity. Here is the chain of reasoning:

1
Expand: \( (1 - \cos t)^2 + \sin^2 t = 1 - 2\cos t + \cos^2 t + \sin^2 t = 2 - 2\cos t \)
2
Half-angle identity: \( 1 - \cos t = 2\sin^2(t/2) \), so \( 2 - 2\cos t = 4\sin^2(t/2) \)
3
Take the root: \( \sqrt{4\sin^2(t/2)} = 2|\sin(t/2)| \). On \( [0, 2\pi] \), \( \sin(t/2) \geq 0 \), so the absolute value drops.

A rolling wheel traces a cycloid

A brass wheel with one marked rim point stands on a rail. Three equal-radius positions in the diagram trace a cycloid from(0,0) through(π,2) to(2π,0).A radius-one wheel rolling without slipping traces x = t − sin t and y = 1 − cos t. Over 0 ≤ t ≤ 2π, the wheel advances 2π units while its rim point travels an arch of length 8.

A rolling wheel traces a cycloid

A brass wheel with one marked rim point stands on a rail. Three equal-radius positions in the diagram trace a cycloid from(0,0) through(π,2) to(2π,0).A radius-one wheel rolling without slipping traces x = t − sin t and y = 1 − cos t. Over 0 ≤ t ≤ 2π, the wheel advances 2π units while its rim point travels an arch of length 8.Open full diagram

Example 4: Astroid

Find the total arc length of the astroid: \( x = \cos^3 t \), \( y = \sin^3 t \), for \( 0 \leq t \leq 2\pi \).

x y

Common Mistakes to Avoid

✕ Forgetting the Square Root
You integrate \( \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 \) directly without taking \( \sqrt{\phantom{x}} \). This gives the wrong units and value.
✕ Wrong Bounds for \( t \)
Using \( x \)-bounds instead of \( t \)-bounds, or choosing a range that retraces the curve (e.g., \( 0 \) to \( 4\pi \) for a circle).
✕ Dropping Absolute Values
Writing \( \sqrt{\sin^2 t} = \sin t \) instead of \( |\sin t| \). This matters when \( \sin t < 0 \).

Summary of Arc Length Formulas

Curve Type Formula Variable
Parametric \( \displaystyle\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\, dt \) \( t \)
Rectangular \( \displaystyle\int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2}\, dx \) \( x \)
Polar \( \displaystyle\int_\alpha^\beta \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\, d\theta \) \( \theta \)

Worked practice: Practice Problems, Problem 4 — applied

Work through this original lesson problem, then reveal the solution and illustrations.

A particle moves along a path defined by x(t) = e^t cos t and y(t) = e^t sin t for 0 ≤ t ≤ 2π. (a) Show that (dx/dt)² + (dy/dt)² = 2e^(2t). (b) Set up the integral for the arc length of the path. (c) Evaluate the integral exactly. (d) Interpret the result in terms of the spiral's geometry.

Show answer and explanation

Point 1 (Derivatives): dx/dt = e^t cos t − e^t sin t = e^t(cos t − sin t). dy/dt = e^t sin t + e^t cos t = e^t(sin t + cos t). Point 2 (Simplification): (dx/dt)² = e^(2t)(cos t − sin t)² = e^(2t)(cos²t − 2sin t cos t + sin²t) = e^(2t)(1 − sin 2t). (dy/dt)² = e^(2t)(sin t + cos t)² = e^(2t)(1 + sin 2t). Sum: e^(2t)(1 − sin 2t + 1 + sin 2t) = 2e^(2t). ✓ Point 3 (Integral setup): L = ∫₀²π √(2e^(2t)) dt = ∫₀²π √2 · e^t dt. Point 4 (Evaluation): L = √2 · [e^t]₀²π = √2(e^(2π) − 1). Point 5 (Interpretation): The curve is a logarithmic (equiangular) spiral. The speed e^t√2 increases exponentially, so the spiral winds outward with each revolution covering dramatically more distance than the previous one. The factor e^(2π) ≈ 535.5 reflects this exponential growth over one full revolution.

Exponential spiral x=e^t cos t, y=e^t sin t from t=0 to 2π, starting at (1,0). Speed is √2 e^t and path length is √2(e^(2π)−1).
The full spiral and enlarged start are a single package. Local chord triangles explain accumulated path length. Endpoint displacement has magnitude e^(2π)−1, whereas path length is √2(e^(2π)−1).
Enlarged spiral for 0≤t≤π on equal coordinate scales, plus two local chords at t=0,0.2,0.4 with horizontal and vertical component legs. Chord lengths approximate arc lengths.
Local chord triangles approximate arc length; equal Δt=0.2 steps lie on x=e^t cos t, y=e^t sin t. Deliver with the full-spiral view.

Question 1 of 5

What is the arc length of the curve \( x = 5\cos t \), \( y = 5\sin t \) for \( 0 \leq t \leq \pi \)?

Question 2 of 5

For \( x = t^2 \), \( y = t^3 \), what expression goes under the radical in the arc length integral?

Question 3 of 5

The arc length of one arch of the cycloid \( x = t - \sin t \), \( y = 1 - \cos t \) from \( t = 0 \) to \( t = 2\pi \) is:

Question 4 of 5

Which condition is not required for the parametric arc length formula to apply on \( [a, b] \)?

Question 5 of 5

For \( x = e^t \cos t \), \( y = e^t \sin t \) on \( [0, \pi] \), which simplification is correct after computing \( \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 \)?