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Calculus I / Applications of Integration

Arc Length and Surface Area

How long is a curve y=f(x)y = f(x) from x=ax = a to x=bx = b? The answer is the arc length L=∫ab1+[f′(x)]2 dxL = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx, derived from the Pythagorean theorem applied to infinitesimal chord lengths. The arc-length element ds=1+[f′(x)]2 dxds = \sqrt{1 + [f'(x)]^2}\,dx reappears throughout mathematics: as the line element in curvilinear coordinates, as the time element for a particle travelling at speed ∣v(t)∣|v(t)|, and as the basis for surface area of revolution. In finance, the arc-length element appears in the realised quadratic variation of a price path, making this lesson a geometric precursor to the stochastic calculus notion of path variation.
Learning objectives. By the end of this note you should be able to:
  • derive and apply the arc-length formula L=∫ab1+(f′)2 dxL = \int_a^b\sqrt{1+(f')^2}\,dx;
  • set up arc-length integrals integrating with respect to yy when that is easier;
  • compute the surface area of a solid of revolution.

In this lesson

  1. The arc-length formula
  2. Arc length integrating with respect to yy
  3. Surface area of revolution
  4. Finance context: path variation and arc length

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