After placing the image, draw a rectangle over the desired portion of the image, then select both and make a clipping mask. With clipping mask, a portion of the image is made visible and the rest is hidden. But if you really want to get rid of the hidden portion of the image, here's a trick.
- Place the clipped image in a blank area of the document.
- From the Transparency panel, apply a blending mode to it - say Multiply. As there is nothing beneath, no change in appearance would happen.
- Now use Object > Flatten Transparency on the image. Remember to choose the right ppi for the raster.
And here you go, hidden portions of the image get removed. But remember this is a trick.