Find the area of the surface defined by the function f(x,y) over the region defined below.  Click here to see a brief discussion of the development of the surface area integral formula used in this problem along with pictures (Powerpoint Presentation).  Click on the picture at the right to see an enlargement.  Click here to see an animation showing a variety of views.  Quicktime version

Here is the Maple worksheet that creates pictures that zoom in on the point of tangency shown above.  QT version

Click here to see an animation showing different views of four approximating pieces of planes.  QT version

Click here to see an animation showing different views of the fifteen approximating pieces of planes.  QT version

Surface Area Example

Find the surface area of the solid of intersection of the cylinders whose equations are x² + z² = 1 and y² + z² = 1.

DPGraphPicture of the intersecting cylinders

DPGraphPicture of the surface

DPGraphPicture of the portion of the surface above the region at the right

DPGraphPicture of the surface and the plane tangent to the surface at the point (0.6,0.2,0.8) over the region specified below.

Zoom-in-out on the plane tangent to the surface x² + z² = 1 at the point given below:

   Use the scrollbar and activate a.  If you change a.maximum to 100 you will not be able to tell the difference between the surface and the tangent plane when a = 100.

A picture of the triangular region over which the double integral will be iterated is shown at the right above.  

          DPGraph view from the top of the portion of the surface over the region shown above

Click the picture to enlarge.

Here is a Maple Worksheet that graphs the cylinders and computes the desired surface area.

DPGraph Picture   You can use the scrollbar to vary b from 0 to 16.