surface integral & surface area

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