Arc Length

The arc length of a curve measures its actual length along the curve (not the straight-line distance between endpoints). Calculating arc length uses integration and is a key topic in AQA Further Mathematics.

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1 Arc Length in Cartesian Form

Derivation

Consider a small element of a curve y = f(x). Between x and x + delta x, the curve moves:

• Horizontally by delta x
• Vertically by delta y

The length of this small element is approximately:

delta s = sqrt((delta x)^2 + (delta y)^2) = sqrt(1 + (delta y / delta x)^2) * delta x

In the limit as delta x -> 0:

ds = sqrt(1 + (dy/dx)^2) dx

The total arc length from x = a to x = b is:

s = integral from a to b of sqrt(1 + (dy/dx)^2) dx

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2 Worked Example 1 — Parabola

Find the arc length of y = x^(3/2) from x = 0 to x = 4.

Solution:

dy/dx = (3/2) x^(1/2)

(dy/dx)^2 = (9/4) x

s = integral from 0 to 4 of sqrt(1 + 9x/4) dx