Why must the total area under a probability density function equal one?

In probability theory, the total area under a probability density function must equal 1 because it represents the total probability of all possible outcomes in the sample space.12

By definition, a random variable accounts for every potential outcome of an event. In the axiomatic foundation of probability, a probability measure is defined such that the probability of the entire sample space $S$ is $P(S)=1$. For a continuous random variable, the probability of an outcome falling within a specific range is calculated by integrating the density function over that interval. Since the integral over the entire range of possible outcomes must cover the full sample space, the total area must sum to 1 to remain consistent with the fundamental requirement that the probability of all certain events is unity.312