Let f(x,y) be a differentiable function. As we have seen, z=f(x,y) defines a surface in xyz-space. In some applications, it necessary to know the surface area of the surface above some region R in the xy-plane. See the figure.
The formula for the surface area is
This is a double integral.
Example
What is the surface area of the plane z=2x+3y above the rectangle with -1<=x<=2 and 0<=y<=2? In this case f_x=2 and f_y=3. Applying the above formula, the surface area S is given by
Since, the region of integration R is a rectangle and the integrand is continuous, the value of the integral is independent of the order of integration. It can be shown that S=6*sqrt(14).
Example
Find the surface area of the part of the paraboloid z=16-x^2-y^2 that lies above the xy plane (see the figure below).
The region R in the xy-plane is the disk 0<=x^2+y^2<=16 (disk or radius 4 centered at the origin).
For this problem, f_x=-2x and f_y=-2y. Hence, the surface area S is given by
Since R is a disk, it convenient to convert the above integral into polar coordinates. The disk R satisfies 0<=r<=4 and 0<=theta<=2*pi. In addition,
The surface area is given by the integral
Both iterated integrals above can be computed in a straightforward manner. The final answer is
Derivation of the Surface Area Formula
It is instructive to derive the surface area formula. We start by assuming that the surface is the plane:
Consider a part of the plane above a rectangle in the xy-plane with x_0<=x<=x_0+dx and y_0<=y<=y_0+dy, as shown in the figure below.
Let u be the vector from point 0 to point 1 and v be the vector from point 0 to point 2. The area of the plane above the rectangle R is
Given the formula for the plane, point 0 is (x_0,y_0,z_0), point 1 is (x_0+dx,y_0,z_0+adx), point 2 is (x_0,y_0+dy,z_0+bdy). Hence,
Taking the cross product, we have
The area is
Note dxdy is the area of the rectangle in the xy plane.
For a general surface z=f(x,y), we can approximate the area of the surface over the small rectangle in the figure above by the tangent plane through (x_0,y_0,z_0). The equation of the tangent plane is
This last equation is the same as the equation for the plane with a replaced by the x derivative and b replaced by the y derivative. Hence, the area is
In the case that the region R is not a rectangle we replace dxdy by dA, the area of a general infinitesimal region containing (x_0,y_0).
[Vector Calculus Home] [Math 254 Home] [Math 255 Home] [Notation] [References]
Copyright © 1996 Department of Mathematics, Oregon State University
If you have questions or comments, don't hestitate to contact us.