To really crop an image in Illustrator, so that you can discard extraneous portions and keep only what you want, do the following:
1. Draw a rectangle (filled) where you want to crop (as you know, short of using a clipping mask, a raster image is always rectangular) and set its opacity to 0%;
2. Select raster image and rectangle, go to Object>Flatten Transparency, slide the quality slider to 100%, and make sure the Alpha Channel option is UNchecked;
3. Deselect all, and select the portion you want to keep, release the clipping mask and extract;
4. Discard unused portions.