Finding Arc Lengths of Curves Given by Parametric Equations
Accumulate parameter speed and distinguish a single curve trace from repeated traversals.
College Board Topic 9.3 · CED p. 168 (PDF p. 173) · CED effective Fall 2020; project course outline 2026–27.
Over a tiny time interval, the point changes by approximately . The Pythagorean theorem gives a tiny traveled length of approximately . Adding those lengths and taking a limit gives for continuously differentiable coordinate functions.
The integrand is speed along the parameterized path, so it is nonnegative. If the parameter covers the curve once, this is its geometric arc length. If it retraces a segment or makes another lap, the integral counts every traversal. State the parameter interval and check tracing before calling the result the length of the distinct curve.
A calculator is useful when the speed has no convenient antiderivative. Enter the speed, not the slope, and report the defining integral as well as the decimal. Arc length has distance units when coordinates have distance units; a slope is dimensionless when both axes use the same unit.
Let's work through it
- For , , , find , . Why: Arc length uses coordinate rates.
- Speed is . Why: The magnitude of the displacement rate is the traveled-length rate.
- Integrate: . Why: Accumulate speed over the whole parameter interval.
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Question 1
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Retrieval 1
Worked answer and reason
sin²t+cos²t=1.
Retrieval 2
Worked answer and reason
The parameter makes two laps, so the integral counts two traversals.
Catch the tempting mistake
- Arc length is ∫(dy/dx)dt.
- Integrate the magnitude of the coordinate derivative vector; a slope measures a different quantity.
Calculator workspace and limits
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Scientific calculator
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Put it to work
- Unit 9 problem 7 · No calculator · analytic
- Unit 9 problem 8 · No calculator · multiple
- Unit 9 problem 9 · No calculator · verbal
For the teacher
- Compare a single lap, two laps, and a forward/backward line segment. Ask for the counted object before the integral.
Glossary and deeper reading · 6 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.
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Differentiability
Differentiability means that a function has a finite derivative at the point under discussion. Differentiability at a point implies continuity there, but continuity alone does not imply differentiability. For example, |x| is continuous at 0 but its left and right tangent slopes differ, so it is not differentiable there. Corners, cusps, vertical tangents, and discontinuities can prevent a finite derivative.
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Integrand
The integrand is the function or expression being integrated. The differential identifies the variable of integration. For example, in the integral of x² with respect to x, the integrand is x². Bounds belong to the integral’s interval rather than to the integrand itself.
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Arc length
Arc length measures the distance along a curve over a specified parameter interval. For a smooth graph y=f(x), its length is the integral of √(1+[f′(x)]²) with respect to x. For example, the graph y=x from x=0 to x=1 has length √2. Parametric curves instead integrate the magnitude of their velocity, accounting for how the curve is traced.
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Antiderivative
An antiderivative of a function is another function whose derivative equals the original function on an interval. Two antiderivatives on the same interval differ by a constant. For example, x² is an antiderivative of 2x, and x²+7 is another. An antiderivative is a function, whereas a definite integral is an accumulated value.
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Integral
Integration can mean accumulating a function over an interval or finding an antiderivative. A definite integral gives a number, while an indefinite integral represents a family of antiderivatives. For example, integrating 2x indefinitely gives x²+C, and integrating it from 0 to 2 gives 4. The Fundamental Theorem of Calculus connects these two operations under its hypotheses.