Suppose we are given a function f and we want to calculate the surface area of the function f rotated around a given line. The calculation of surface area of revolution is related to the arc length calculation.
If the function f is a straight line, other methods such as surface area formulas for cylinders and conical frustra can be used. However, if f is not linear, an integration technique must be used.
Recall the formula for the lateral surface area of a conical frustum:
where r is the average radius and l is the slant height of the frustum.
For y=f(x) and , we divide [a,b] into subintervals with equal width Δx and endpoints . We map each point to a conical frustum of width Δx and lateral surface area .
We can estimate the surface area of revolution with the sum
As we divide [a,b] into smaller and smaller pieces, the estimate gives a better value for the surface area.
Definition (Surface of Revolution)
The surface area of revolution of the curve y=f(x) about a line for is defined to be
The Surface Area Formula
Suppose f is a continuous function on the interval [a,b] and r(x) represents the distance from f(x) to the axis of rotation. Then the lateral surface area of revolution about a line is given by
And in Leibniz notation
= = =
As and , we know two things:
1. the average radius of each conical frustum approaches a single value
2. the slant height of each conical frustum equals an infitesmal segment of arc length
From the arc length formula discussed in the previous section, we know that
Because of the definition of an integral , we can simplify the sigma operation to an integral.
Or if f is in terms of y on the interval [c,d]