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15.5 - Section 16.5 Applications of Integration Using...

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Section 16.5 Applications of Integration “Using Double Integrals to find Surface Area and other Applications” In Calculus 2, we used integrals to find the surface area of a surface of revolution. Though this method could be used to find the surface area of the sphere, a paraboloid and some other surfaces, the fact that it only works for surfaces which are revolutions about some axis means that it cannot be used to find the areas of most surfaces. In this section, we use double integrals to determine a formula for surface area of any surface (provided the defining equations are nice enough). 1. Surface Area Suppose that S is a surface which is the graph of some function z = f ( x, y ) which has continuous partial derivatives and suppose we want to find the surface area of f ( x, y ) over some region R in the plane (see illustration). We proceed as follows. ( i ) We break up the region R into small rectangles with side lengths of Δ x and Δ y . We ignore any rectangles which are not fully contained in R (we will be taking a sum as Δ x and Δ y go to zero). ( ii ) Let the point (¯ x i , ¯ y j ) denote the point in he bottom left hand corner of each rectangle (we could choose any point, but we choose this one for convenience).

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