Finding the Area of a 3D Surface with Parallelograms

f(x,y) =

2

x

+

2

y

parallelograms (per dimension)

2

surface opacity (red)

1

parallelogram opacity (blue)

1

red area: 7.446

blue area: 6.9282

This Demonstration shows a sequence of Riemann-like approximations of the area of a surface in three-space. The area of a surface can be approximated by using an array of parallelograms that are tangent to the surface at their centers. The magnitude of the cross product of two vectors that describe adjacent sides of a parallelogram yields the area of the parallelogram, and finding the sum of all of these areas results in a good approximation of the area of the surface. As the number of parallelograms increases, the approximation becomes more accurate, as you can see by using the slider. The double integral is constructed using the same idea of summing approximations taken over successively smaller regions of the

x-y

plane.