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Volume Between Two Surfaces
Our first application is to the volume between two surfaces. DEFINITION Let f(x, y) ≤ g(x, y) for (x, y) in D and let E be the set of all points in space such that (x, y) is in D, f(x, y) ≤ z ≤ g(x, y). The volume of E is V is called the volume over D between the surfaces z = f(x, y) and z = g(x, y) (Figure 12.3.4). Figure 12.3.4: Volume between two surfaces JUSTIFICATION The part ΔE of the solid E over an element of area ΔD is a rectangular solid with base ΔA and height g(x, y)  f(x, y), except that the top and bottom surfaces are curved (Figure 12.3.5). Therefore the volume of ΔE is ΔV ≈ (g(x, y)  f(x, y)) ΔA (compared to ΔA). By the Infinite Sum Theorem, Figure 12.3.5


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