AP® Calculus BC review sheet from Aim for Five (aimforfive.com/calc-bc/units/8/8-13)
Unit 8 · Topic 8.13
BC only8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled
BC only. The length of a smooth curve y = f(x) from x = a to x = b is ∫ₐᵇ √(1 + [f′(x)]²) dx. The formula adds up tiny hypotenuses along the curve, and most arc length integrals need a calculator.
Key terms
- arc length
- smooth curve
- derivative
- Pythagorean theorem
Where the formula comes from
This topic is tested only on the BC exam.
Zoom in on a smooth curve until a tiny piece looks straight. Over a small run Δx, the curve rises Δy, so that piece is a hypotenuse with length √((Δx)² + (Δy)²). Factor out Δx: √(1 + (Δy/Δx)²)·Δx. As the pieces shrink, Δy/Δx becomes f′(x), and adding them up gives
L = ∫ₐᵇ √(1 + [f′(x)]²) dx.
The curve must be smooth: f′ must be continuous on [a, b]. No corners or vertical tangents inside the interval.
Other forms
- For a curve x = g(y) from y = c to y = d: L = ∫ from c to d of √(1 + [g′(y)]²) dy.
- For a parametric curve (x(t), y(t)): L = ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt (9.3). That's the same hypotenuse idea, with t as the variable.
- For a particle moving along a curve, that parametric integral is the total distance it travels (9.6).
Exact vs. calculator answers
The square root usually makes arc length integrals impossible to do by hand. Exam questions without a calculator use special curves where 1 + (f′)² is a perfect square or simplifies nicely, such as y = (2/3)x^(3/2). With a calculator, set up the integral and evaluate it numerically.
A quick sanity check: the arc length is always at least the straight-line distance between the endpoints. If your answer is shorter, something's wrong.
Why smoothness matters
At a corner, the derivative jumps, and at a vertical tangent it blows up. The formula can still be applied on pieces where f′ is continuous: split the curve at the problem point and add the lengths. In context, arc length answers questions like how long a cable, a road or a piece of trim along a curved edge is. The units are the same as the units on the axes, like meters.
Setting it up step by step
- Find f′(x).
- Square it, add 1 and put it under a square root.
- Use the x-limits of the piece of curve you're measuring.
- Evaluate exactly if 1 + (f′)² simplifies; otherwise use a calculator.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
A curve built to work by hand
Find the length of y = (2/3)x^(3/2) from x = 0 to x = 3.
Show the solutionHide the solution
- Step 1: y′ = (2/3)(3/2)x^(1/2) = √x.
- Step 2: 1 + (y′)² = 1 + x.
- Step 3: L = ∫₀³ √(1 + x) dx = (2/3)(1 + x)^(3/2) from 0 to 3 = (2/3)(8 − 1) = 14/3.
Answer: 14/3 ≈ 4.667
- Example 2Calculator allowed
Calculator: a parabola
Find the length of y = x² from x = 0 to x = 2.
Show the solutionHide the solution
- Step 1: y′ = 2x, so 1 + (y′)² = 1 + 4x².
- Step 2: L = ∫₀² √(1 + 4x²) dx ≈ 4.647.
- Step 3: Check: the straight line from (0, 0) to (2, 4) has length √20 ≈ 4.472. The curve is a bit longer, as it should be.
Answer: About 4.647
- Example 3
Trap: square the derivative, not the function
A student writes the length of y = x² on [0, 2] as ∫₀² √(1 + x⁴) dx. What's wrong?
Show the solutionHide the solution
- Step 1: The formula uses [f′(x)]², not [f(x)]².
- Step 2: Here f′(x) = 2x, so the integrand is √(1 + 4x²), not √(1 + x⁴).
- Step 3: Another common slip is leaving out the 1: ∫₀² √(4x²) dx = ∫₀² 2x dx = 4, which is shorter than the straight-line distance, so it can't be right.
Answer: The correct integral is ∫₀² √(1 + 4x²) dx ≈ 4.647.
Common mistakes
- Squaring f(x) instead of f′(x).
- Forgetting the 1 under the square root.
- Forgetting to square the derivative, as in √(1 + f′(x)). The formula needs [f′(x)]² under the root.
- Using arc length on a curve that has a corner or vertical tangent inside the interval without splitting it.
On the exam
- Arc length shows up in BC multiple choice, often as “which integral gives the length.” Look for √(1 + (f′)²).
- On free response, it can be a quick part of a region question. Write the integral, then the calculator value.
Connected topics
Videos
Check yourself
4 questions on 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled. Pick an answer to see if you got it, and why.
Which of the following gives the length of the graph of y = x³ from x = 0 to x = 2?
What is the length of the graph of y = (2/3)x^(3/2) from x = 0 to x = 3?
The function f is defined for x ≥ 1 and f′(x) = √(x⁴ − 1). What is the length of the graph of f from x = 1 to x = 3?
What is the length of the graph of y = ln x from x = 1 to x = 3?
0 of 4 answered