# What's a circular integral?

## Main Question or Discussion Point

"
If we designate the total or internal energy of an isolated or closed system as E, heat as Q and work as W , then the circular integral involving no change in net internal energy is:...
"

What is it?

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Andy Resnick
I think that refers to a path integral- the integral is taken to be along a closed loop, rather than along a segment of a coordinate axis.

jtbell
Mentor
That's what it must be: integral around a closed path. I've never used the term "circular integral" myself, or seen it in any of my textbooks, but a Google search shows that it does appear in some thermodynamics books.

Yes, it was in a chapter about thermodynamics.

My mom said that it was a volume-based integral, or something like that, but it's still a little bit unclear (partially because I only know how to integrate in a straight line)

And here is what it looks like:

$$\oint$$

A circular integral is the integration around a closed path or surface. For example, Gauss's law says that
$${\Phi} = \oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = {1 \over \varepsilon_0} \int_V \rho\ \mathrm{d}V = \frac{Q_A}{\varepsilon_0}$$
where you're integrating around a closed surface (such as a sphere, or any other structure else where you can tell at every point whether you're inside or outside).

Doc Al
Mentor
As others have pointed out, that's usually called a closed path integral (if you are integrating along a line or path).