Surface Area of Revolution

When a curve is rotated about an axis, it sweeps out a surface of revolution. Calculating the area of this surface extends the arc length concept and is an important topic in AQA Further Mathematics.

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1 Surface Area about the x-axis

Derivation

Consider a small element of the curve y = f(x) with arc length ds. When this element is rotated about the x-axis, it sweeps out a thin band (frustum of a cone) with:

• Circumference = 2pi y (the distance from the curve to the x-axis)
• Width = ds

The surface area of this band is approximately:

dS = 2pi y * ds

Since ds = sqrt(1 + (dy/dx)^2) dx, the total surface area is:

S = 2pi * integral from a to b of y * sqrt(1 + (dy/dx)^2) dx

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2 Worked Example 1 — Cone

Find the surface area generated by rotating y = 2x (for 0 <= x <= 3) about the x-axis.

Solution:

dy/dx = 2, so sqrt(1 + 4) = sqrt(5).

S = 2pi * integral from 0 to 3 of 2x * sqrt(5) dx