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Population density is a measure of how many individuals exist per unit area. In the context of this exercise, it refers to the concentration of a bacterium over a certain geographical space described by a function. In mathematical terms, the function provided, \[ f(x, y) = \frac{10000 e^y}{1 + \frac{|x|}{2}} \]expresses the population density of bacteria in the x-y plane. Here, the variable \(x\) gauges horizontal position, whereas \(y\) measures vertical position.The base of the formula, \(10000 e^y\), suggests that the population density changes exponentially with \(y\). As \(y\) increases, the density grows exponentially, suggesting that areas higher on the y-axis have more bacteria.This density depends inversely on \(|x|\), meaning sections further from the y-axis experience lower density, while regions closer to the y-axis have a denser population. Breaking it down:- \(10000 e^y\): Increases density with higher \(y\)- \(1 + \frac{|x|}{2}\): Decreases density with increasing \(|x|\) proximityUnderstanding population density functions like this is imperative in various fields such as ecology, epidemiology, and urban planning, facilitating the calculation of total populations over specified regions.

A rectangular region in this context refers to a specific part of the plane where we calculate the population density to derive total population. The boundaries of this region are defined by:- \(-5 \leq x \leq 5\)- \(-2 \leq y \leq 0\)The x-coordinates range from -5 to 5, creating a horizontal span across the plane, while y ranges from -2 to 0, signifying the vertical bounds. Together, these x and y boundaries form a rectangle.Utilizing a rectangle simplifies integration because it ensures consistent and straightforward limits for double integration.Evaluating functions specifically over rectangular regions is common in calculus because it assists in defining limits of integration:

• A rectangle allows straightforward integration using Cartesian coordinates.
• Helps in evaluating functions with respect to both dimensions easily.

Understanding the rectangular confines assures accurate assessment of the total area when integrating population density functions, providing results exclusive to only those plotted dimensions in the plane.

Integral calculus is the branch of mathematics concerned with the accumulation of quantities, such as areas under curves, and finding totals, like sums of changes. For this exercise, we utilize a double integral to measure the total population over a specific rectangular region.The problem at hand requires us to calculate:\[P = \int_{-2}^{0} \int_{-5}^{5} \frac{10000 e^y}{1 + \frac{|x|}{2}} \, dx \, dy\]The double integral accumulates the total bacteria population over the defined boundaries. Calculating it involves two integrations:

• First with respect to \(x\), treating \(y\) as a constant.
• Then, over \(y\), summing up results from the \(x\)-integration.

This methodology is foundational in fields requiring the total measurement of distributed quantities across geometrical shapes, leveraging the strengths of integration to solve complex real-world problems.

Symmetry plays a vital role in simplifying double and single integrals, saving calculative effort. Specifically in this exercise, we observe symmetry about \(x = 0\). The function \[ \frac{1}{1 + \frac{|x|}{2}} \]is symmetric about the y-axis.When integrating from \(-5\) to \(5\), the function remains the same, whether \(x\) is positive or negative. This symmetry allows us to:

• Split the integral because effects cancel out each other over symmetrical regions.
• Negate the computation necessity over negative and positive separately.

Utilizing symmetry reduces the complexity of calculations substantially, enabling straightforward results. By acknowledging symmetric properties, we managed to succinctly compute challenging integrals easily applicable in practical situations.