AP® Calculus BC review sheet from Aim for Five (aimforfive.com/calc-bc/units/9/9-3)
Unit 9 · Topic 9.3
BC only9.3 Finding Arc Lengths of Curves Given by Parametric Equations
BC only. The length of a parametric curve from t = a to t = b is ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt. If the curve is the path of a moving particle, the same integral is the total distance it travels.
Key terms
- arc length
- parametric curve
- distance traveled
- smooth curve
The formula
This whole unit is BC only. Over a tiny time interval Δt, the point moves about Δx horizontally and Δy vertically, so it covers a distance of about √((Δx)² + (Δy)²). Factor out Δt: √((Δx/Δt)² + (Δy/Δt)²)·Δt. Add these up and take the limit:
L = ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt.
The derivatives dx/dt and dy/dt should be continuous on [a, b] (the curve is smooth).
Connection to arc length in 8.13
If the curve is y = f(x), you can use x itself as the parameter: x = t, y = f(t). Then dx/dt = 1 and dy/dt = f′(t), and the formula becomes ∫ √(1 + [f′(t)]²) dt, exactly the 8.13 formula. The parametric version is the general one.
Length of the curve vs. distance traveled
If a particle traces part of its path more than once, the integral counts every pass. For x = 3 cos t, y = 3 sin t from 0 to 4π, the integral gives 12π, which is the distance traveled (two laps), even though the circle itself is only 6π long.
So when a question asks for the length of a curve, make sure your t-interval traces it exactly once. When it asks for total distance traveled by a particle, use the full time interval it's given.
Exact vs. calculator
Set your calculator to radian mode before integrating anything with trig functions of t. Most of these integrals require a calculator. The no-calculator versions are designed so that (dx/dt)² + (dy/dt)² is a perfect square, or simplifies with an identity such as sin² t + cos² t = 1. Watch for those patterns.
Speed is the integrand
The quantity √((dx/dt)² + (dy/dt)²) is the speed of a point moving along the curve (9.6). So arc length is the integral of speed, just as in Unit 8 the distance traveled along a line is the integral of |v(t)|. If you know the speed at every moment, you know how far the point goes.
A quick reasonableness check: the length must be at least the straight-line distance between the starting and ending points. If the curve is close to a straight line, the two numbers should be close too.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
A perfect-square integrand
Find the length of the curve x = t², y = (2/3)t³ for 0 ≤ t ≤ √3.
Show the solutionHide the solution
- Step 1: dx/dt = 2t and dy/dt = 2t².
- Step 2: (dx/dt)² + (dy/dt)² = 4t² + 4t⁴ = 4t²(1 + t²).
- Step 3: For t ≥ 0, √(4t²(1 + t²)) = 2t√(1 + t²).
- Step 4: L = ∫ from 0 to √3 of 2t√(1 + t²) dt. Let u = 1 + t², du = 2t dt, with u running from 1 to 4.
- Step 5: L = ∫₁⁴ √u du = (2/3)(4^(3/2) − 1) = (2/3)(7) = 14/3.
Answer: 14/3 ≈ 4.667
- Example 2Calculator allowed
Calculator length
Find the length of the curve x = t³ − 3t, y = t² for 0 ≤ t ≤ 2.
Show the solutionHide the solution
- Step 1: dx/dt = 3t² − 3 and dy/dt = 2t.
- Step 2: L = ∫₀² √((3t² − 3)² + (2t)²) dt.
- Step 3: Calculator: L ≈ 7.605.
Answer: About 7.605
- Example 3
Trap: tracing a curve twice
Find the length of the circle x = 3 cos t, y = 3 sin t. A student integrates from t = 0 to t = 4π. What do they get?
Show the solutionHide the solution
- Step 1: dx/dt = −3 sin t, dy/dt = 3 cos t, so the integrand is √(9 sin² t + 9 cos² t) = 3.
- Step 2: From 0 to 2π (one trip around), L = 3(2π) = 6π. That matches the circumference 2π(3).
- Step 3: From 0 to 4π, the integral gives 12π, because the point goes around twice. That's a distance traveled, not the circle's length.
Answer: The circle's length is 6π; integrating over 0 to 4π gives 12π, which double-counts.
Common mistakes
- Forgetting to square dx/dt or dy/dt, or adding them before squaring.
- Using a t-interval that traces part of the curve twice when the question asks for the curve's length.
- Writing √((dx/dt)² + (dy/dt)²) with x-limits instead of t-limits.
- Dropping the absolute value when simplifying √(4t²): it's 2|t|, which equals 2t only for t ≥ 0.
On the exam
- On calculator free response, you'll often be asked for the distance traveled by a particle on a time interval. Write the integral, then the number.
- Multiple-choice questions may ask which integral gives a parametric arc length. Check that both derivatives are squared and the limits are t-values.
Connected topics
Videos
Check yourself
4 questions on 9.3 Finding Arc Lengths of Curves Given by Parametric Equations. Pick an answer to see if you got it, and why.
Which of the following gives the length of the path described by x = t² and y = t³ from t = 0 to t = 2?
What is the length of the curve defined by x = t² and y = (2/3)t³ for 0 ≤ t ≤ √3?
A curve is defined by x = ln t and y = sin t. What is the length of the curve from t = 1 to t = 3?
What is the length of the curve defined by x = eᵗ cos t and y = eᵗ sin t for 0 ≤ t ≤ 1?
0 of 4 answered