Calculate Population Density Using Integration

Hey guys! Let's dive into an interesting problem involving population density and how we can use some cool math to figure it out. We're going to explore how to calculate the population within a specific region of a city using a given population density model. It's like being a city planner, but with more equations! This involves understanding population density, mathematical models, and the magic of integration.

Understanding the Population Density Model

In this scenario, the population density of a city is approximated by the model:

f(x, y) = 2,000e^{-0.01(x^2 + y^2)}

This formula might look a bit intimidating at first, but let's break it down. Here, f(x, y) represents the population density at a specific point (x, y) in the city. The variables x and y are measured in miles, giving us a spatial context. The region we're interested in is defined by x² + y² ≤ 121. This inequality describes a circular area centered at the origin (0, 0) with a radius of 11 miles (since √121 = 11). The constant 2,000 is a scaling factor, indicating the peak population density at the city center, and the exponential term e^{-0.01(x2 + y^2)} models how the population density decreases as we move away from the center. The beauty of this model lies in its ability to provide a smooth, continuous representation of population distribution, making it amenable to mathematical analysis.

The exponential decay is crucial here; it mirrors the real-world phenomenon of populations thinning out as distance from a city center increases. Think about it: the heart of a city is bustling, but the outskirts are more residential and less densely populated. The 0.01 coefficient in the exponent controls the rate of this decay. A smaller coefficient would mean a slower decay, implying that the population density remains higher further from the center. Conversely, a larger coefficient would lead to a more rapid drop-off in density. Understanding these parameters allows us to model different types of cities – some with a sharp central peak and others with a more gradual population spread.

The region x² + y² ≤ 121 is also super important because it defines the boundaries within which we're calculating the population. This circular region simplifies the integration process, as we'll see later. However, it's a simplification of real-world city shapes, which are rarely perfect circles. Real-world applications might involve more complex regions defined by irregular boundaries, requiring more advanced mathematical techniques to handle. For our purposes though, the circular region is a great starting point.

So, to recap, this model gives us a way to estimate how many people live in different parts of our city. It’s dense at the center and gradually thins out as you move outwards, all within a circle with an 11-mile radius. The challenge now is to use this density function to find the total population within this region. And that, my friends, is where integration steps in!

Integrating the Density Function: The Key to Finding Total Population

To find the total population, we need to integrate the density function over the given region. Think of integration as summing up all the tiny bits of population across the entire area. In mathematical terms, this means we need to evaluate a double integral. Double integrals are the tool we use to find volumes under surfaces, or in our case, to sum up a density function over a two-dimensional region. The integral we need to solve looks something like this:

∬ f(x, y) dA

Where f(x, y) is our population density function, and dA represents an infinitesimal area element. The double integral symbol (∬) tells us we’re integrating over a two-dimensional area. The key here is to set up this integral correctly, taking into account the region we’re integrating over.

Because our region is a circle, it's much easier to use polar coordinates. Polar coordinates use the distance from the origin (r) and the angle (θ) to define a point, instead of x and y. This simplifies the integral because the circular region x² + y² ≤ 121 becomes r ≤ 11 in polar coordinates, and the area element dA transforms to r dr dθ. This transformation makes our lives significantly easier, guys!

To convert our function to polar coordinates, we use the relationships x = r cos θ and y = r sin θ. Thus, x² + y² = r², and our density function becomes:

f(r, θ) = 2,000e^{-0.01r^2}

Now, our double integral in polar coordinates looks like this:

∬ 2,000e^{-0.01r^2} r dr dθ

The limits of integration are crucial. Since we're integrating over the entire circle, r will go from 0 to 11 (the radius), and θ will go from 0 to 2π (a full circle). This gives us the following definite integral:

∫[0 to 2π] ∫[0 to 11] 2,000e^{-0.01r^2} r dr dθ

This integral represents the total population within the 11-mile radius. Solving this integral will give us a numerical value, which is our estimated population. Don't worry, we'll walk through the steps to solve this in the next section. For now, the key takeaway is that converting to polar coordinates simplifies the problem immensely, turning a potentially messy integral into a manageable one. This is a common trick in multivariable calculus, and it's incredibly powerful when dealing with circular or radial symmetry.

Solving the Integral: Step-by-Step

Alright, let's get our hands dirty and solve this integral. We've already set up the integral in polar coordinates:

∫[0 to 2π] ∫[0 to 11] 2,000e^{-0.01r^2} r dr dθ

First, we'll tackle the inner integral with respect to r:

∫[0 to 11] 2,000e^{-0.01r^2} r dr

This looks like a perfect candidate for a u-substitution. Let's set u = -0.01r². Then, the derivative du = -0.02r dr. We can rearrange this to get r dr = du / -0.02. Also, we need to change the limits of integration. When r = 0, u = 0. When r = 11, u = -0.01(11)² = -1.21. So, our integral becomes:

∫[0 to -1.21] 2,000e^u (du / -0.02)

We can simplify this by pulling out the constants:

-100,000 ∫[0 to -1.21] e^u du

Now, the integral of e^u is simply e^u. So, we have:

-100,000 [e^u] from 0 to -1.21

Evaluating this gives us:

-100,000 (e^{-1.21} - e^0)

-100,000 (e^{-1.21} - 1)

This is the result of the inner integral. Now, we need to plug this into the outer integral with respect to θ:

∫[0 to 2π] -100,000 (e^{-1.21} - 1) dθ

Notice that the expression inside the integral is a constant with respect to θ. So, we can pull it out:

-100,000 (e^{-1.21} - 1) ∫[0 to 2π] dθ

The integral of dθ from 0 to 2π is simply 2π. Therefore, our final result is:

-100,000 (e^{-1.21} - 1) * 2π

Now, let’s calculate this value. e^{-1.21} is approximately 0.298. Plugging this in, we get:

-100,000 (0.298 - 1) * 2π

-100,000 (-0.702) * 2π

70,200 * 2π

140,400π

This is approximately 441,030.5. So, the estimated population within the 11-mile radius is about 441,031 people. That's a lot of people!

Interpreting the Result and Real-World Implications

Okay, so we've crunched the numbers and found that the estimated population within the 11-mile radius is approximately 441,031 people. But what does this number really tell us? And how can it be used in the real world? Let's break it down.

Firstly, it's important to remember that this is an approximation based on our model. The model f(x, y) = 2,000e^{-0.01(x2 + y^2)} is a simplified representation of the city's population density. Real-world population distributions are often more complex, influenced by factors like zoning laws, natural barriers (rivers, mountains), and the presence of industrial or commercial areas. So, while our result gives us a good estimate, it's not a perfectly precise figure. Think of it as a high-level overview rather than a detailed street-by-street count.

However, even with its limitations, this kind of calculation has significant practical applications. City planners can use population estimates to make informed decisions about resource allocation. For instance, knowing the approximate number of residents in an area helps determine the need for schools, hospitals, public transportation, and other essential services. Imagine trying to plan the construction of a new hospital without knowing how many people it needs to serve! Our calculation provides a valuable starting point for such planning processes.

Moreover, these population estimates can be used for infrastructure development. If a city is experiencing population growth in a particular area, it might need to expand its road network, upgrade its water and sewage systems, or build new power plants. Our calculation can help identify areas of high population density and growth, allowing city officials to anticipate future infrastructure needs and allocate resources accordingly. It’s like having a crystal ball that gives you a glimpse into the city’s future!

Businesses also benefit from population density information. A company looking to open a new store, restaurant, or service center will want to know where its potential customers are located. Areas with high population density are generally more attractive for businesses, as they offer a larger customer base. Our calculation can help businesses identify promising locations and make strategic decisions about where to invest.

Furthermore, demographic analysis relies heavily on population estimates. By combining our calculation with other data sources, such as census information and surveys, we can gain a deeper understanding of the city's population characteristics. This includes factors like age distribution, income levels, and household size. This information is crucial for a wide range of purposes, from social policy development to marketing campaigns.

In conclusion, while our population estimate is based on a mathematical model, it has real-world implications for city planning, infrastructure development, business decisions, and demographic analysis. It’s a powerful tool that helps us understand and manage the complexities of urban life. And who knows, maybe you guys will use similar models in your future careers!

Conclusion

So, guys, we've taken a journey from a seemingly complex population density model to a concrete estimate of the city's population. We've seen how mathematical models can be used to represent real-world phenomena, and how integration can be used to extract meaningful information from these models. This process, while involving some mathematical heavy lifting, ultimately provides valuable insights for urban planning, resource allocation, and business strategy. The key takeaways here are the power of mathematical modeling and the versatility of calculus in solving practical problems. Keep exploring, keep questioning, and who knows what other exciting applications of mathematics you'll discover! Remember, math isn't just about numbers and equations; it's about understanding the world around us.