Surface Area with Parametrizations

Proposition (Surface Area Defined Parametrically) Let be a surface defined parametrically by



on the region in the -plane, and assume that is smooth in the sense that and are continuous with on Then the surface area, is given by



The quantity is called the fundamental cross product.

    Proof. Suppose a surface is defined parametrically by the vector function  



for parameters and Let be a region in the -plane on which and as well as their partial derivatives with respect to and are continuous. The partial derivatives of are given by



Suppose the region is subdivided into cells. Consider a typical rectangle in this partition, dimension and , where and are small. If we project this rectangle onto the surface we obtain a curvilinear parallelogram with adjacent sides and The area of this rectangle is approximated by





By taking an appropriate limit, we find the surface to be a double integral.

Example (Surface Area Defined Parametrically) Find the area of the surface given parametrically by the equation



for
    
    Solution. We have,  



Therefore,


Using polar coordinates, the surface area is











Example (Surface Area Defined Parametrically) Find the surface area of the helicoid which is given parametrically by  



for and

    Solution. We have,





and



Therefore,



So the surface area is

Example (Surface Area of a Torus)  Find the surface area of the torus which is given parametrically by  



for and
    
    Solution. We have,





and





Therefore,



So the surface area is

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Cite this as:
Surface Area With Parametrizations
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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