How to export to JPG only what fits inside the actual page?

Open your Illustrator file in Photoshop and save it in JPG format

torput,

You may perform the normally destructive deed:

1) Create a rectangle covering the desired part (stroke, no fill makes it easier);

2) Select both and Object>Clipping Mask>Make;

3) In the Transparency palette dropdown, select anything bu normal;

4) Object>Flatten Transparency (no need to change options).

4) should get rid of the outlying parts for you.

Select your mask and double click on the Crop Area tool, or if in CS4 the Art Board tool from the options pull down select Fit Crop Area to Selected Art. Now try to export your jpg.