A student wants to find the length of the curve y = x² from x = 0 to x = 1. She sets up the integral ∫₀¹ x² dx and gets 1/3. What error has she made?
AShe computed the area under the curve rather than arc length; the correct setup is ∫₀¹ √(1 + (2x)²) dx
BShe used the wrong limits of integration; she should integrate from 0 to f(1) = 1
CShe forgot to multiply by 2π, which converts area integrals to length
DShe should integrate |f(x)| rather than f(x) to get path length
Integrating f(x) directly gives the signed area between the curve and the x-axis — not the curve's length. Arc length requires summing the lengths of infinitesimal hypotenuses √(dx² + dy²), which simplifies to ∫√(1 + [f′(x)]²) dx. For y = x², f′(x) = 2x, so the arc length integral is ∫₀¹ √(1 + 4x²) dx — quite different from ∫₀¹ x² dx.
Question 2 Multiple Choice
In the arc length formula L = ∫ₐᵇ √(1 + [f′(x)]²) dx, where does the '1 +' inside the square root come from?
AIt represents the horizontal component dx of each infinitesimal segment, via the Pythagorean theorem applied to the tiny right triangle with legs dx and dy
BIt is a normalizing constant that ensures the integral converges for all continuous functions
CIt shifts the integrand upward to prevent the square root from taking a negative value
DIt represents the constant of integration absorbed into the formula during derivation
Each infinitesimal segment of the curve has horizontal run dx and vertical rise dy = f′(x)dx. By the Pythagorean theorem, the segment length is √(dx² + dy²) = √(dx² + [f′(x)]² dx²) = √(1 + [f′(x)]²) dx. The '1' comes from dx² / dx² = 1 after factoring out dx. It is always present because even a nearly horizontal curve still has a horizontal component.
Question 3 True / False
Most arc length integrals that arise from natural functions cannot be evaluated in elementary closed form.
TTrue
FFalse
Answer: True
True. For most functions — y = sin(x), y = x³, y = eˣ — the expression √(1 + [f′(x)]²) does not have an elementary antiderivative. The 'nice' examples in calculus textbooks (like y = x^(3/2)) are specially engineered so the algebra simplifies. Arc length is a natural context where numerical integration is often the only practical option.
Question 4 True / False
The arc length formula is derived by summing the vertical changes dy along the curve from x = a to x = b.
TTrue
FFalse
Answer: False
False. Arc length sums the lengths of infinitesimal hypotenuses — √(dx² + dy²) — not just the vertical changes. Summing only dy would give the total net vertical displacement (f(b) − f(a)), not the path length. The Pythagorean theorem is essential: both horizontal and vertical components must be included.
Question 5 Short Answer
Explain in your own words why the arc length formula involves a square root, and what geometric idea that square root represents.
Think about your answer, then reveal below.
Model answer: The curve is cut into infinitely many tiny segments. Each segment is approximately a straight line with a horizontal component dx and a vertical component dy = f′(x) dx — a tiny right triangle. By the Pythagorean theorem, the length of that segment (the hypotenuse) is √(dx² + dy²). Factoring out dx gives √(1 + [f′(x)]²) dx. The square root is the hypotenuse of an infinitesimal right triangle; arc length is the integral (sum) of infinitely many such hypotenuses.
Recognizing the Pythagorean theorem as the source of the square root is what lets you adapt the formula to parametric curves and surfaces of revolution — the derivation is always the same geometric idea in a slightly different setting.