1. An image processing apparatus for cutting out an image corresponding to an original from an image obtained by reading the original using a reading device, the image processing apparatus comprising: a memory configured to store images; an input unit configured to input, in units of band images, an image obtained by reading an original using the reading device; a detection unit configured to analyze the band image input by the input unit and detect an area corresponding to the original within the band image; a storage control unit configured to cause a partial image containing the area detected by the detection unit within the band image to be stored in the memory; and a cut-out unit configured to cut out an image corresponding to the original from the partial image stored in the memory.