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Population density refers to how many people live in a specific area, typically measured per square mile or square kilometer. In our exercise, we have a function describing the population density of a city as a function of distance from the city center. Specifically, the population density is given by the formula: \[120 e^{-0.65 t}\], where \( t \) represents the distance in miles from the city center and the resulting value is in thousands of people per square mile. Understanding this function helps us to see how the population spreads out as you move farther from the city center.

A definite integral calculates the accumulated value or total amount over an interval, and it's crucial in determining quantities like areas, volumes, or, in our case, the number of people within a specific distance. For the given problem, we want to find out how many people live within 5 miles of the city center. This involves setting up the definite integral:\[ \text{Number of people} = \int_{0}^{5}120 e^{-0.65 t} \, dt \]. This integral sums up the population density function from \( t = 0 \) to \( t = 5\), giving us the total population within the 5-mile radius.

To solve the definite integral, we first need the antiderivative, or the function inverse to differentiation. Finding the antiderivative of \(120 e^{-0.65 t}\) involves recognizing that it takes the form associated with exponential functions. Using calculus rules, the antiderivative is: \[ \int 120 e^{-0.65 t} \, dt = -\frac{120}{0.65} e^{-0.65 t}+ C \]. Simplifying, we get: \[ -184.615 e^{-0.65 t} + C \]. This step is vital as it transforms the problem into one where we can substitute the limits of the integral to find the total number of people, as explained next.

The given population density function, \(120 e^{-0.65 t}\), exhibits exponential decay, meaning the density decreases as the distance from the city center increases. Exponential decay functions are of the form \( ae^{-kt} \) where \( k > 0 \) determines the rate of decay. Here, the function \( e^{-0.65 t}\) highlights how rapidly the density falls as you move outward. Evaluating the integral from \( t = 0 \) to \( t = 5 \):\[ \int_{0}^{5}120 e^{-0.65 t} \, dt = [-184.615 e^{-0.65 t}]_{0}^{5} = \left(-184.615 e^{-3.25}\right) + 184.615 \]. Calculating this gives approximately 177.462 thousand people. This result shows how exponential decay models can apply to real-life scenarios, like urban population distribution.