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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 DEFINITION The area of a smooth surface z = f(x, y), (x, y) in D is JUSTIFICATION Let S(D_{1}) be the area of the part of the surface with (x, y) in D_{v }S(D_{1}) has the Addition Property, and S(D_{1}) ≥ 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,


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