When a curve y = f(x) for a ≤ x ≤ b is rotated about the y-axis, the surface of revolution is given by

     

In this section, the definition for a surface area generated from revolving a curve, which is described using the parametric equations, around an axis will be introduced.

Surface of Revolution:
Parametric Form

Example: Surface Area of a Sphere

Similar to the concept of an arc length, when a curve is given by the following parametric equations

     x = f(t) and y = g(t) for a ≤ t ≤ b,

the surface area of revolution for the curve revolving around the y-axis is defined as

Consider a sphere with a center at (0,0) and radius of 2. The sphere can be obtained by rotating a semi-circle

     x = 2cos(t) and y = 2sin(t) for 0 ≤ t ≤ π

about the x-axis. According to the definition of the surface of revolution, the surface area of the sphere is given by