Mathfolis

Arc Length

Unit 8 · Applications of Integration

Arc Length (BC Only)

The arc length of a smooth curve y = f(x) on [a, b] is L = ∫√(1 + (dy/dx)²) dx. The key step is simplifying 1 + (f′)² — this often yields a perfect square, enabling a clean square root.

Arc Length Formulas

Curve y = f(x)
L=∫ab1+[f′(x)]2 dxL = \int_a^b \sqrt{1 + \left[f'(x)\right]^2}\,dx
Parametric / vector motion
L=∫t1t2[x′(t)]2+[y′(t)]2 dtL = \int_{t_1}^{t_2}\sqrt{[x'(t)]^2+[y'(t)]^2}\,dt
AP Tip: After computing (dy/dx)², add 1 and look for a perfect square under the radical before integrating — this is the most common simplification.
Type 1

Arc Length of y = f(x)

Differentiate f, square it, add 1, simplify the radical, then integrate.

Example 1
Find the arc length of y=(2/3)x3/2y = (2/3)x^{3/2} on [0, 3].
Example 2
Find the arc length of y = ln(cos x) on [0, π/4].
Type 2

Setting Up Arc Length Integrals

For many arc length problems on the AP exam, full evaluation is difficult — the key skill is correctly setting up the integral.

Example 3
Write (but do not evaluate) the integral for the arc length of y = x² + sin x on [0, π].
Example 4
A particle has position x(t) = t², y(t) = t³. Write the integral for the distance traveled from t = 0 to t = 2.
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