Difference Between Surface Integrals and Surface area using double integrals .

[shenjie]

Hi all, Thanks for response :)
I Dont really understand what is surface integrals ?? and its difference with Surface Area using double integrals. Can anyone help ? thanks a lot...

 

[Vorde]

I'm assuming your familiar with multivariable calculus up to this.

Realize that when you are taking a double or triple integral what you are doing is summing up all the values of a function in a region of space or a region of the plane.
A line integral is what you get if instead of wanting to sum the values in a region, you want to sum them up over a path in space- a line.
A surface integral is what you get if instead of a region or a path, you want to sum up values over a surface.
Now I'll give you some more concrete examples.

Lets say I had a function P(x,y) that described the density of any point (and remember that weight (really mass) is the integral of density).

If I wanted to know how much an infinitely flat circle of radius 1 weighed in this density field, I would take the double integral of the function over the circle of radius one - because the object I'm interested in (the circle) can be regarded as a region.

If I wanted to know how much a ball of radius 1 weighed in this density field, I would take the triple integral of the function over the sphere of radius 1 - because the object I'm interested in (the sphere) can also be regarded as a region.

But let's say I have an infinitely skinny copper wire and I want to find out how much it weighs. I can't take a double or triple integral because I can't describe this wire as a region, so I take the line integral over the path the wire is in and I can get the weight of the wire. The line integral is used because I can't describe the object I want as a region, but I can describe it as a line.

Now let's imagine I have a infinitely skinny piece of paper that is bent in a weird way and I want to know the weight of it. I can't treat it as a region because it isn't flat on the plane like the circle is, and it doesn't follow a path like the wire does. But it does follow a surface, so then I can take the surface integral over the surface that the bent piece of paper is making and then I can know the weight of it.

Line integrals, Surface integrals and Double/Triple integrals are all just extensions of the integral to different dimensions and when integrating over different shapes. Finding the surface area with integrals is just using the properties of integrals to determine what the surface area of a graph would be if it represented a physical shape - it does something completely different than surface integrals.

 

[shenjie]

Thanks a lot. I finally have a vision on this topic :)