Surface Area

To find the area of a surface defined by z = f(x,y) over a region of the xy-coordinate plane we will begin by approximating the area of the surface.  We will do this by partitioning the region of the xy-coordinate plane into n rectangles.  Some of the rectangles in the finite partition of the region might extend outside the region.  That is not happening in the picture shown below since the region in the picture shown is a rectangle.  In each rectangular partition in the xy-coordinate plane a point is being chosen and then the point on the surface with the same x- and y-coordinates is determined.  At that point on the surface the plane tangent to the surface at that point is determined.  This plane is then "trimmed" into a parallelogram directly above the corresponding rectangular partition in the xy-coordinate plane.  The area of this parallelogram could be computed using the magnitude of the cross product of the two vectors shown in red in the picture below on the right.  These two vectors could be approximated as indicated in the analysis below on the left.

Click on the picture to see an enlargement. Click here to zoom in on the tangent plane.

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        Lane Vosbury, Mathematics, Seminole State College   email:  vosburyl@seminolestate.edu

        This page was last updated on 08/21/14          Copyright 2002          webstats