Integrate Over Domain? | General Integration Questions

[Chemmjr18]

I have a question about integration and I hope I can word it correctly. When we integrate over something (i.e. a line, a surface, or volume), what exactly do I integrate over? Is it the domain of that "thing"? I feel like it's something I've been told but I just can't remember. If this is the case visualizing integration would be significantly easier for me . It would all come down to knowing the domain.

 

[Mark44]

I have a question about integration and I hope I can word it correctly. When we integrate over something (i.e. a line, a surface, or volume), what exactly do I integrate over? Is it the domain of that "thing"? I feel like it's something I've been told but I just can't remember. If this is the case visualizing integration would be significantly easier for me . It would all come down to knowing the domain.

The "over" part is an interval or a two-dimensional region or three-dimensional region, and so on.

For this integral -- $\int_0^2 f(x)dx$ -- the integration takes place over the interval [0, 2].
For this integral -- $\int_0^2 \int_0^5 f(x, y)~dy~dx$ -- integration takes place over the rectangular region [0, 2] X [0, 5] in the plane.

 

[Chemmjr18]

For this integral -- $\int_0^2 f(x)dx$ -- the integration takes place over the interval [0, 2].
For this integral -- $\int_0^2 \int_0^5 f(x, y)~dy~dx$ -- integration takes place over the rectangular region [0, 2] X [0, 5] in the plane.

In each of these cases, we're just integrating over the domain of the region?

 

[fresh_42]

In each of these cases, we're just integrating over the domain of the region?

Yes, and the result is the volume between this domain and the surface or line. But one has to keep an eye on zeroes, as it is an oriented volume which can get negative. For example $\int_{-\pi}^\pi \cos(x)\,dx = 0$ although there is definitely an area under this curve, but the one below the $[-\pi,\pi]$ and the one above cancel each other out.

 

[Chemmjr18]

Got it. Thank for your help!

 

[Mark44]

In each of these cases, we're just integrating over the domain of the region?

I prefer to save the word "domain" for the set of possible values for the function. Instead of "domain" I say "interval" or "region."

 

[jtbell]

I prefer to save the word "domain" for the set of possible values for the function. Instead of "domain" I say "interval" or "region."

The term I seem to remember learning a long time ago is "range", but that may apply to the function as a whole, not the restricted portion that one integrates.