"Bowel" (or "bowels") is a collective term for the large and small intestine. The small intestine is the part of the digestive tract that immediately follows the stomach, and it is where the majority of digestion and nutrient absorbtion takes place. The small intestine is followed by the large intestine. This is where water is absorbed into whatever indigestible food matter is left in the system and it is then moved along and elimated as waste, or feces.
In casual or non-medical conversation, "intestines" and "bowels" are often used interchangeably. It is always understood that the speaker was having lower abdominal pain/difficulty whether he describes it as originating in his intestines or his bowels.
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