# Integral with circle in it.

#### PrudensOptimus

what does that sign mean? a integral sign with a circle on it.

#### Ambitwistor

It usually refers to a line integral, which is the integral of a function around a closed loop in the domain of the function. Sometimes it is extended to higher dimensions, e.g. to represent an integral over a closed surface.

#### PrudensOptimus

so this line integral is the same as a normal integral?

#### Ambitwistor

Originally posted by PrudensOptimus
so this line integral is the same as a normal integral?
A line integral is performed not over the entire domain of the integrand, but only over a one-dimensional subspace of the domain (a closed curve).

#### Tom Mattson

Staff Emeritus
Gold Member
Originally posted by Ambitwistor
A line integral is performed not over the entire domain of the integrand, but only over a one-dimensional subspace of the domain (a closed curve).
A "normal" integral need not be performed over the entire domain of the integrand, either.

Originally posted by PrudensOptimus
so this line integral is the same as a normal integral?
By "normal integral" I take you to mean "integral along the x-axis".

A line integral is a generalization of a "normal integral". Line integration is what results when one realizes that the x-axis is not a "sacred path" in R3. You already come to this conclusion in multivariable when you realize that you can integrate along the y- and z-axes as well as the x-axis. But you take this notion further in line integration when you remove the restriction that the path of integration be a straight line.

The so-called "normal integral" is a line integral. The "line" is just the x-axis. If you want the "normal" analog to the closed path integral, then you can integrate your function from a to b, and then add to it the integral of the same function from b to a.

#### HallsofIvy

Homework Helper
I will also point out that a "line integral" is NOT the same as "the integral of a function around a closed loop in the domain of the function".

A line integral is the integral of a function of 2 or more independent variables along a given path between 2 points in the domain of definition. You can write the one-dimensional path in terms of parametric equations in one parameter, write the function, restricted to that path, in terms of that parameter and integrate that in exactly the way you do "normal integrals" (of one variable).

Such a path does not HAVE TO BE closed. If it is, then you have the integral symbolized by the "integral sign with a circle on it".

Everywhere you see it, the symbol $\oint$ may be replaced with $\int$ with no change in meaning. The circle is only there to emphasize the fact that the path (or surface or whatever) you are integrating over is closed.

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