Question

The population density (in people per square mile) for a coastal town can be modeled by \(f(x, y)=\frac{120,000}{(2+x+y)^{3}}\) where \(x\) and \(y\) are measured in miles. What is the population inside the rectangular area defined by the vertices \((0,0)\), \((2,0),(0,2)\), and \((2,2) ?\)

Short answer

The population inside the rectangular area defined by the vertices (0,0), (2,0), (0,2), and (2,2) is approximately 13043.

Step-by-step solution

Understand the problem

We are given a function of two variables, \(x\) and \(y\), which represents the population density of a coastal town. We need to find the population of the town in the region between \((0,0)\), \((2,0)\), \((0,2)\), and \((2,2)\). This can be calculated by integrating the population density function over this region.

Set up the double integral

The double integral will allow us to sum up all the tiny pieces of population over the whole region. The region is a square with sides along the \(x\) and \(y\) axes from 0 to 2. Thus the double integral becomes \[ \int_0^2 \int_0^2 f(x, y) \, dx \, dy = \int_0^2 \int_0^2 \frac{120,000}{(2+x+y)^{3}} \, dx \, dy \].

Solve the inside integral

First, we solve the integral with respect to \(x\). Hold \(y\) constant and integrate \( \frac{120,000}{(2+x+y)^{3}} \) from 0 to 2. This becomes a regular integral with the new variable \(u = 2 + x + y\) and \(du = dx\). Using substitution, we get \( \int_0^2 \frac{120,000}{u^3} \, du = -\frac{60,000}{u^2}\) evaluated from 0 to 2.

Solve the outside integral

Next, solve the integral with respect to \(y\). The integrated value with respect to \(x\) is \(-\frac{60,000}{(2+y)^2}\) from 0 to 2. The new variable is \(v = 2 + y\) and \(dv = dy\), so we get \( \int_0^2 -\frac{60,000}{v^2} \, dv = 60,000v^{-1} \) evaluated from 0 to 2.

Evaluate the limits of the integral

Finally, plug in the limits of integration to get the exact value of the population inside the specific region. After performing the evaluation, we have estimated the population to be approximately 13043.

Key concepts

Population Density

Population density is a way to understand how crowded a place is. In the case of a coastal town, like the one in the exercise, population density talks about how many people live in each square mile.
This is important when you want to know how busy or spacious an area might feel.
The population density function given in the exercise is \( f(x, y)=\frac{120,000}{(2+x+y)^{3}} \). It takes two inputs, \(x\) and \(y\), measured in miles.
These inputs tell the exact point in the town you're looking at.
Using this function, you can find how crowded any given spot in the town is.

• The bigger the values of \(x\) and \(y\), the smaller the population density since \((2 + x + y)^3\) gets larger.
• This means people spread out more as you move further from the starting point at \((0,0)\).

Understanding population density not only helps in planning and resource allocation for the town but also helps in calculating the total population when you know the area you're interested in.

Calculus of Two Variables

The calculus of two variables involves functions with two inputs, like \( x \) and \( y \), rather than just one.
This area of calculus is crucial when dealing with scenarios involving spatial calculations, such as the area or volume in a physical region.
In our problem, the population density function is a function of two variables.
This means it considers both the horizontal measurement \(x\) and the vertical measurement \(y\) to understand how population density changes across different points.
This type of function can be used in fields like physics, economics, and environmental studies where such spatial dimensions are essential.

• With calculus of two variables, you can integrate functions over areas to find things like total population, mass, or charge.
• It allows for a detailed analysis of changes that occur over an area instead of just along a line.

The calculus of two variables extends traditional calculus by allowing for the integration over areas, capturing more complex and realistic models of the world.

Rectangular Region Integration

Integrating over a rectangular region is a common method in double integration.
It involves integrating a function over a defined area that forms a rectangle.
In the exercise, this rectangle is outlined by the vertices \((0,0)\), \((2,0)\), \((0,2)\), and \((2,2)\). The double integral setup is \[ \int_0^2 \int_0^2 \frac{120,000}{(2+x+y)^{3}} \, dx \, dy \]This expression sums the population density from every tiny piece within the square.
This process captures the entire population spread across the area.

• The first integral deals with one dimension (let's say \(x\)) while holding the other constant.
• The second integral then deals with the other dimension (\(y\)).

Evaluating these integrals, starting from the inside out, provides the total accumulation, which in our example, denotes the total population within the defined rectangle.
This technique is vital in various fields such as physics and engineering, helping to solve problems related to area-specific inquiries.