Problem 41 - Thomas Calculus

Question

Bacterium population If \(f(x, y)=\left(10,000 e^{y}\right) /(1+|x| / 2)\) represents the "population density" of a certain bacterium on the \(x y\) -plane, where \(x\) and \(y\) are measured in centimeters, find the total population of bacteria within the rectangle \(-5 \leq x \leq 5\) and \(-2 \leq y \leq 0\)

Step-by-Step Solution

Answer

Total population is \(40000 \ln\left(\frac{7}{2}\right) \cdot (1 - \frac{1}{e^2})\).

Step 1: Recognize Integral Setup

The total population of the bacterium is the integral of the population density function over the given region. The density function is given by \[ f(x, y) = \frac{10000 e^y}{1 + \frac{|x|}{2}} \]We need to integrate this function over the rectangle defined by \[-5 \leq x \leq 5\] and \[-2 \leq y \leq 0\] to find the total population.

Step 2: Set Up the Double Integral

The total population can be calculated using a double integral over the given rectangular region:\[P = \int_{-2}^{0} \int_{-5}^{5} \frac{10000 e^y}{1 + \frac{|x|}{2}} \, dx \, dy\]This expression sums up the population density function over the specified range of \(x\) and \(y\).

Step 3: Simplify the Integral with Respect to x

First, we integrate the function with respect to \(x\). Since \(|x|\) is symmetrical about zero, it simplifies our integration. For the integral with respect to \(x\), note that it's evaluated separately for positive and negative values, but the symmetry cancels out the need to differentiate. You can write the simplified function as:\[\int_{-5}^{5} \frac{1}{1 + \frac{|x|}{2}} \, dx = \int_{-5}^{5} \frac{2}{2 + |x|} \, dx\]This integral needs to be evaluated next.

Step 4: Evaluate the Inner Integral

Calculate the integral:\[\int_{-5}^{5} \frac{2}{2 + |x|} \, dx\]Given the symmetry and properties of the function, for \(-5 \leq x < 0\) and \(0 \leq x \leq 5\), split and evaluate these integrals:\[\int_{-5}^{0} \frac{2}{2 - x} \, dx + \int_{0}^{5} \frac{2}{2 + x} \, dx\]Evaluate these integrals to find contributions from each side.

Step 5: Solve the Inner Integral for Two Parts

Let's calculate these parts separately:1. Evaluate \[\int_{-5}^{0} \frac{2}{2 - x} \, dx = 2 \ln|2+x|\Bigg|_{-5}^{0}, \] giving \[2 (\ln(2) - \ln(7)).\]2. Evaluate \[\int_{0}^{5} \frac{2}{2 + x} \, dx = 2 \ln|2+x|\Bigg|_{0}^{5}, \] which is \[2 (\ln(7) - \ln(2)).\]

Step 6: Combine and Conclude Inner Integral Result

Adding these results together gives the inner integral over \(-5\) to \(5\):\[2 \ln(7) - 2 \ln(2) + 2 \ln(7) - 2 \ln(2) = 4 \ln(7/2)\]This is the result of integrating with respect to \(x\).

Step 7: Evaluate the Outer Integral with Respect to y

Substitute the result from Step 6 back into the outer integral:\[\int_{-2}^{0} 10000 e^y \cdot 4 \ln\left(\frac{7}{2}\right) \, dy\]Factor out constants and evaluate:\[40000 \ln\left(\frac{7}{2}\right) \cdot \int_{-2}^{0} e^y \, dy = 40000 \ln\left(\frac{7}{2}\right) \cdot (e^0 - e^{-2})\] which simplifies to:\[40000 \ln\left(\frac{7}{2}\right) \cdot (1 - \frac{1}{e^2})\].

Step 8: Evaluate and Simplify

Calculate the actual value. Assuming computations are accurate:\[40000 \ln\left(\frac{7}{2}\right) \cdot (1 - \frac{1}{e^2})\]Substitute the known constants (using calculator for numerical calculation) and simplify to yield an approximate total population value.

Key Concepts

Population Density

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.

Rectangular Region

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

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 in Integration

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.