Example 4: Surface Of a Sphere

Derive the formula A = 4πr² for the area of the surface of a sphere of radius r.

When the portion of the circle x² + y² = r² in the first quadrant is rotated about the y-axis it will form a hemisphere of radius r (Figure 6.4.12). The surface of the sphere has twice the area of this hemisphere.

Figure 6.4.12

It is simpler to take y as the independent variable, so the curve has the equation

x =, 0 < y < r.

Then

This derivative is undefined at y = 0. To get around this difficulty we let 0 < a < r and divide the surface into the two parts shown in Figure 6.4.13, the surface B generated by the curve from y = 0 to y = a and the surface C generated by the curve from y = a to y = r.

Figure 6.4.13

The area of C is

We could find the area of B by taking x as the independent variable. However, it is simpler to let a be an infinitesimal e. Then B is an infinitely thin ring-shaped surface, so its area is infinitesimal. Therefore the hemisphere has area

½A = B + C ≈ 0 + 2πr(r - ε) ≈ 2πr² ,

so

½A = 2πr²,

and the sphere has area

A = 4πr².