CalcRef

Applications of Integrals · Arc Length & Surface Area

Arc Length

L=∫ab1+[f′(x)]2 dxL = \int_a^b \sqrt{1 + [f'(x)]^2}\, dx

The length of a curve y = f(x) from x = a to x = b.

Variables

SymbolNameUnit
aLeft bound—
bRight bound—

Worked examples

Find the arc length of y = x^(3/2) from x = 0 to x = 4.
  1. f'(x) = (3/2)x^(1/2). [f']² = (9/4)x
  2. L = ∫₀⁴ √(1 + 9x/4) dx. Let u = 1+9x/4, du = 9/4 dx
  3. = (4/9)·(2/3)[u^(3/2)]₁¹⁰ = (8/27)(10√10 - 1)

Answer: (8/27)(10√10 - 1) ≈ 9.073

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