Area Of A Smooth Surface

In Chapter 6 we were able to find the area of a surface of revolution by a single integral. To find the area of a smooth surface in general (Figure 13.5.1), we need a double integral. We call a function f(x, y), or a surface z = f(x, y), smooth if both partial derivatives of f are continuous.

Figure 13.5.1

JUSTIFICATION

Let S(D₁) be the area of the part of the surface with (x, y) in D_v S(D₁) has the Addition Property, and S(D₁) ≥ 0. Consider the piece of the surface ΔS above an element of area ΔD (Figure 13.5.2). ΔS is infinitely close to the piece of the tangent plane above ΔD, which is a parallelogram with sides

Figure 13.5.2

The quickest way to find the area of this parallelogram is to use the vector product formula (Section 10.4, Problem 39),

Area = |U x V|.

Then

Therefore

(compared to Δx Δy),

and by the Infinite Sum Theorem,