# What exactly is a surface integral?

I'm learning surface integrals right now and I don't think I fully understand what they are. What exactly do surface integrals represent? Is it volume? The basis for surface integrals seems just like a standard double integral using some surface.

Like any integral it is adding up a continuous thing over an area. In context this can mean a variety of different concepts. It could be a volume if the function is representing height - this is probably a good analogy to start with - but it could be far harder to conceptualize depending on the situation.

Examples:

In fluid dynamics if we had different pressures over the surface of an object then the integral of the pressure over that area would give the total force pushing on the object.

In optics if we shone a light on a wall then the integral of the intensity of the light over the surface area of the wall would relate to the energy produced by the light.

• My textbook calls it "the double integral analog of the line integral". Can you explain this to me? – Victor Mao Dec 1 '15 at 6:20
• What is your understanding of a line integral? How does that differ to your understanding of a 1D integral? – Ian Miller Dec 1 '15 at 6:22
• Well a line integral measures the area under a line in 3D space by relating some curve with a function – Victor Mao Dec 1 '15 at 6:23
• Hmm seems like you probably want to develop a different way to think about line integrals first then extend that to surface integrals. In a 1D integral we can always think of it as the limit of the Riemann sum rather than as an area. You are adding up the values of the function at different points and averaging it by the number of points then making the number of points go to infinity (and finally times by the length). This idea can now apply to a line integral - we are adding up lots of values along the line and dividing by the number of points on the line. – Ian Miller Dec 1 '15 at 6:29
• In a surface integral we can imagine some quantity (defined by the function) having different values all over the surface. If we break the surface into small areas we can add up the average value in each area. This is our Riemann sum equivalent. We can then make the areas get smaller and smaller approaching a limit. – Ian Miller Dec 1 '15 at 6:31