I saw somewhere triple line integrals

[LSE1234]

I saw somewhere triple line integrals, 3 integrals with a circle(the symbol) , may you tell me exactly what are called, to find & study them ?

 

[HallsofIvy]

Those are NOT "line integrals".

\oint d\sigma is an integral over a closed path.

\skew{57} {\skew{15}\subset \supset} \iint \mbox{ } dS is an integral over a closed surface.

<br /> \skew{57} {\skew{15}\subset \supset} \iiint dV\mbox{ }<br /> is an integral over a closed region in space. If you are working in 3 dimensions to begin with, ALL regions are "closed" so the circle is not necessary.

 

[LSE1234]

first is line integral, second...?... third ...?... .
Where I may find problems and how to solve 2nd and 3rd kind of ?

 

[LSE1234]

integral over a closed region in space PART OF USED FOR calculating volume or...?

 

[slider142]

integral over a closed region in space PART OF USED FOR calculating volume or...?

The double integral with a circle over it is called a closed surface integral. It is commonly used to integrate a vector field defined over a closed surface (simply attach a vector to each point on the surface, this is a vector field. Ie., a head of hair or the flux of a fluid through a cross-section, or electromagnetic flux out of a sphere). If you instead integrate just the area form dA over the surface, you get the surface area.
Similarly, the triple integral with the circle over it integrates vector fields over closed volumes, but these are not as easily visualized.
The first courses where one may encounter extensive use of these objects are vector calculus and electrodynamics.

 

[LSE1234]

In CALCULUS II (undergraduate) these are get coveraged ?