Integrating underneath a curve (or a probability ditribution) will give you the area under that curve. The area under the curve between two values on the x axis will give you the probability of getting a value between them. Most probability distributions are "higher" in the "middle" and "lower" toward the "tails" - this indicates that values closer to the middle are more likely to occur, and values toward the tail are less likely.

For a probability distribution, it is a requirement that integrating under the curve from negative infinity to positive infinity must sum to 1, since the sum of the probabilities of all events in a probability distribution must sum to 1.

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calculus and probability.