derivation of second formula for surface integration with respect to area

In this entry, we shall consider how to compute area integrals when the surface is given as the graph of a functionsMathworldPlanetmath and present another derivation of the formula for area integration in this case.

Suppose that g is a function of two variables and that the surface S is the graph of g:


To evaluate Sd2A, we shall begin by subdividing the xy plane into a fine grid. Corresponding to each of the squares of the grid, we shall have a small portion of the surface. As before, we shall assume that, by choosing the grid spacing fine enough, we can ensure that these small pieces are approximately flat, and can be approximated by a portion of the tangent plane to the surface.

Let us consider one of these small rectanglesMathworldPlanetmath into which the xy plane has been subdivided and one of the small portions of surface which lies above it (or rather the portion of tangent plane which approximates it). Now the area of the rectangle is dxdy and the area of the portion of surface is, by definition, d2A.

Now, we use a fact about projections. If a figure F1, located in a plane P1 projects down to a figure F2 in a plane P2, then


where θ is the dihedral angleMathworldPlanetmath between P1 and P2. In our case, P1 is the tangent plane, P2 is the xy plane, F1 is the bit of tangent plane which approximates the portion of surface, and F2 is the rectangle in the tangent plane, and θ is the angle between the tangent plane and the xy plane. Hence, in our case, the formula reads


To finish this derivation, we need to figure out the cosine of the angle between the tangent plane and the xy plane. The tangent plane is described by the equation


and the xy plane is, of course, described by the equation


Then, the intersection of these two planes is given by the line


The plane perpendicularMathworldPlanetmathPlanetmath to this line is given by


Since the angle between two planes is defined as the angle between the angle between the lines perpendicular to the intersection of the planes, we will obtain the cosine of the angle by considering the right triangleMathworldPlanetmath with vertices at


Dividing adjacentPlanetmathPlanetmath by hypotenuseMathworldPlanetmath, we find that


Having computed cosθ, we may now finish our derivation. Substituting into the formula for d2A, we obtain


and, hence,


Let us note that this formula is consistent with the formula we derived earlier — if we set u=x, v=y, and z=g(x,y), then our previous formula for surface integrals reduces to this one.

Title derivation of second formula for surface integration with respect to area
Canonical name DerivationOfSecondFormulaForSurfaceIntegrationWithRespectToArea
Date of creation 2013-03-22 15:07:26
Last modified on 2013-03-22 15:07:26
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 5
Author rspuzio (6075)
Entry type Derivation
Classification msc 28A75