# surface integral & surface area

i would like to know the relationship between surface integral and surface area.

how are they actually related???

Thanks a lot!!!!!

plz give some examples~~~ and it would be great if u can explain more clearly with the formulas~~

### 1 個解答

• 最愛解答

In 3 dimensions, You can have two type of surface areas

1. it bounds a volume which means it is closed.

To work out the Surface area(S. A.), you need to know the shape of the volume and all the planes on the volume

e.g. a cube is bounded by 6 surfaces and you know the way to work out the area of each surface.

e.g. a sphere. The formula is S.A. = 4*Pi*r^2

2. it is open and is called a curve(plane).

In this case, you need to know the regions of the S.A.

If it is not a flat plan, it is always hard to be calculated

and here it comes the Surface Integral

Normally, we express a surface as a function of x and y, i.e. z=f(x,y)

We would like to express it as a vector function:

r=xi+yj+zk

and we want to have 2 variables instead of 3, so let x=u, y=v

then z=f(u,v)

and r becomes

r=ui+vj+f(u,v)k (note that x=u, y=v is not necessary, it can be in other forms)

then we differentiate r with respect to u and v

and we get ru and rv

|ru x rv| is a parallelogram

This area approximates the area of a small part of the surface

Adding these small areas(In the sense of the Riemann Integral)

We get the area of the surface:

A(s)=int(int(|ru x rv|)du)dv in a Domain which is the region of the surface

Besides working out S.A., Surface Integral can work out the integration of a function on a plain.

The formula will become int(G(x,y,z))ds where G is the function we want to integrate and s is the surface we want to integrate on.

It can be written as I = int(int( G( x(u,v),y(u,v),z(u,v) )*|ru x rv| )du)dv

I hope that is enough for you.

It is quite hard for you to learn it on your own, so I suggest you wait until you get a place in university.

Good luck

資料來源： My lecture notes