Problem 2 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A city is in the shape of a rectangle 4 miles wide by 6 miles long. A river runs through the middle of the city, parallel to the 6 mile-long sides. People prefer to live nearer the water, so the density of people is given by \(\rho(x)=10,000-800 x\) people per square mile, where \(x\) is the distance from the river. (You may ignore the width of the river in this problem.) (a) Show in a sketch how you will need to slice up the region. (b) What is the area of the \(i\) th slice? (c) What is the approximate population in the \(i\) th slice? (d) Write a Riemann sum to estimate the total population of the city. (e) Calculate the exact population by taking the limit of the Riemann sum and evaluating the resulting definite integral.

Step-by-Step Solution

Answer

The exact total population of the city is calculated using definite integral, it is given by evaluating \(\int_{-2}^{2} (10,000-800 x) * 6 dx\).

Step 1: Sketch the Region

Visualize and sketch a rectangle representing the city. Since the river divides the city into two, the city will have two halves 2 miles wide each. Use the 'x' direction to mark the distance from the river. Identify slices (strips) perpendicular to the x-axis.

Step 2: Calculate the Area of the i th Slice

The area of each strip (slice) can be calculated as \(width \times height\), where ‘width’ is \( \Delta x\) (a small change in x distance) and 'height' is the length of the city - 6 miles. So the area is \(A_{i}= 6 \Delta x\) square miles.

Step 3: Determine the Population in the i th Slice

The approximate population in the \(i\)th slice can be calculated by multiplying the area with the population density function. So, it is \(pop_{i} = (10,000-800 x_i)(6\ \Delta x)\). Therefore, \(pop_{i}\) is nearly \( (60,000 - 4800 x_i)\ \Delta x\) people.

Step 4: Estimate Total Population with Riemann Sum

The total population can be estimated by summing up the population of all these slices (a Riemann sum). This can be expressed as \(Total_{pop} \approx \Sigma pop_{i} = \Sigma (60,000 - 4800 x_i) \Delta x\), where the sum is over all slices.

Step 5: Calculate Exact Population with Definite Integral

A more precise value for the population can be computed by taking the limit as \( \Delta x \) becomes infinitesimally small, leading to the integral \(\int_{a}^{b} \rho(x) * 6 dx\), where \( a = -2\) and \( b = 2\). So, the total population would thus be \(\int_{-2}^{2} (10,000-800 x) * 6 dx\).

Key Concepts

Riemann sum

A Riemann sum is an essential concept in calculus that helps in approximating the total area under a curve. Think of it as adding up the areas of several narrow rectangles beneath a curve. Each rectangle's width (\( \Delta x \)) is a small portion of the x-axis, and the height is determined by the value of the function at a specific point within that width.

In the context of calculating population in the city, we used Riemann sums to split the city into thin slices parallel to the river. For each slice, we estimated the population based on its distance from the river, which influenced its density. By summing up the populations of all these slices, you get a close estimate of the total population.

• Width of slice: \( \Delta x \)
• Height of slice: Determined by the density function \( \rho(x) \)
• Sum: The collection of populations from all slices

The Riemann sum formula we used was: \( Total_{pop} \approx \Sigma (60,000 - 4800 x_i) \Delta x \). This process translates the problem into manageable pieces for easier calculation.

population density

Population density in this scenario represents how many people live in a square mile of the city, depending on their distance from the river. It's given by the function \( \rho(x) = 10,000 - 800x \), which means that the density decreases as you move away from the river.

Why would density change with distance from the river? People tend to prefer living nearer the water, so population concentration tends to be higher closer to the river. The equation \( \rho(x) \) quantifies this behavior. Let's break it down:

• \(10,000\): The density of people right at the riverbank, the maximum value.
• -800x: The decrement in density for each mile of distance from the river.

For every mile you move away from the river, 800 fewer people per square mile are expected to live there. Understanding this helps in modeling the city's population distribution accurately.

definite integral calculation

The definite integral is a powerful tool in calculus used to find exact values over an interval. When you need a precise calculation, you take the limit as the width of each slice approaches zero. This transforms your Riemann sum into a definite integral.

In this exercise, calculating the total population via a definite integral involved integrating the product of the population density function and the width of the city along the interval from where the city starts to where it ends on the x-axis. The definite integral in this case was:\[ \int_{-2}^{2} (10,000 - 800x) * 6 \, dx \]

• Limits of Integration: From \(-2\) to \(2\), reflecting the city's layout from one side of the river to the other.
• Integrand: The function \((10,000 - 800x) * 6\) representing the integrated slices.

This process precisely sums up infinitely small contributions across the entire city, yielding the exact population. When solving, you perform the integration and then evaluate the result to find the accurate total population.