An example in Surface Area

 

A fun fact

 

          A sphere is cut on the surface into rings as in the diagram. If the vertical height of each of the rings remains the same, say k units. Believe it or not - the surface area of any one of the rings remains constant.

 

          Let x2 + y2 = r2 be the circle as in the diagram. We concentrate on the positive-x part of the function. The [0, r] on the x-axis is cut into n equal intervals. (Here are 5.):

 

          The width of the ring is . Then the function is rotated around the x-axis by 4 right angles.

 

 

  The surface area of the k-th ring, formed by rotating Ik around x-axis, is given by:

         

Now,  x2 + y2 = r2 , therefore

         

Substitute (2) in (1), we can get:

         

                 

Here Sk is independent of k (there is no k in the result). Therefore Sk is a constant.

 

 

          As there are n rings in positive x direction and n rings in the negative x direction.

The total surface area of all rings is