derivation of second formula for surface integration with respect to area

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

functions

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

:

z

=

g

⁢

(

x

,

y

)

To evaluate

∫

S

d

2

⁢

A

, we shall begin by subdividing the

x

⁢

y

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

rectangles

into which the

x

⁢

y

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

d

⁢

x

⁢

d

⁢

y

and the area of the portion of surface is, by definition,

d

2

⁢

A

.

Now, we use a fact about

projections

. If a figure

F

1

, located in a plane

P

1

projects

down to a figure

F

2

in a plane

P

2

, then

area

⁢

(

F

2

)

=

area

⁢

(

F

1

)

⁢

cos

⁡

θ

,

where

θ

is the

dihedral angle

between

P

1

and

P

2

. In our case,

P

1

is the tangent plane,

P

2

is the

x

⁢

y

plane,

F

1

is the bit of tangent plane which approximates the portion of surface, and

F

2

is the rectangle in the tangent plane, and

θ

is the angle between the tangent plane and the

x

⁢

y

plane. Hence, in our case, the formula reads

d

⁢

x

⁢

d

⁢

y

=

cos

⁡

θ

⁢

d

2

⁢

A

.

To finish this derivation, we need to figure out the cosine of the angle between the tangent plane and the

x

⁢

y

plane. The tangent plane is described by the equation

z

=

∂

⁡

g

∂

⁡

x

⁢

x

+

∂

⁡

g

∂

⁡

y

⁢

y

and the

x

⁢

y

plane is, of course, described by the equation

z

=

0

.

Then, the

intersection

of these two planes is given by the line

0

=

z

=

∂

⁡

g

∂

⁡

x

⁢

x

+

∂

⁡

g

∂

⁡

y

⁢

y

.

The plane

perpendicular

to this line is given by

∂

⁡

g

∂

⁡

x

⁢

y

=

∂

⁡

g

∂

⁡

y

⁢

x

.

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 triangle

with vertices at

(

0

,

0

,

0

)

,

(

∂

⁡

g

∂

⁡

x

,

∂

⁡

g

∂

⁡

y

,

0

)

,

and

(

∂

⁡

g

∂

⁡

x

,

∂

⁡

g

∂

⁡

y

,

(

∂

⁡

g

∂

⁡

x

)

2

+

(

∂

⁡

g

∂

⁡

y

)

2

)

.

Dividing

adjacent

by

hypotenuse

, we find that

cos

⁡

θ

=

(

∂

⁡

g

∂

⁡

x

)

2

+

(

∂

⁡

g

∂

⁡

y

)

2

(

∂

⁡

g

∂

⁡

x

)

2

+

(

∂

⁡

g

∂

⁡

y

)

2

+

(

(

∂

⁡

g

∂

⁡

x

)

2

+

(

∂

⁡

g

∂

⁡

y

)

2

)

2

=

1

1

+

(

∂

⁡

g

∂

⁡

x

)

2

+

(

∂

⁡

g

∂

⁡

y

)

2

.

Having computed

cos

⁡

θ

, we may now finish our derivation. Substituting into the formula for

d

2

⁢

A

, we obtain

d

2

⁢

A

=

1

+

(

∂

⁡

g

∂

⁡

x

)

2

+

(

∂

⁡

g

∂

⁡

y

)

2

⁢

d

⁢

x

⁢

d

⁢

y

and, hence,

∫

S

f

⁢

(

x

,

y

)

⁢

d

2

⁢

A

=

∫

f

⁢

(

x

,

y

)

⁢

1

+

(

∂

⁡

g

∂

⁡

x

)

2

+

(

∂

⁡

g

∂

⁡

y

)

2

⁢

𝑑

x

⁢

𝑑

y

.

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

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