Problem 26 - Thomas Calculus

Question

Regional population If \(f ( x , y ) = 100 ( y + 1 )\) represents the population density of a planar region on Earth, where \(x\) and \(y\) are measured in miles, find the number of people in the region bounded by the curves \(x = y ^ { 2 }\) and \(x = 2 y - y ^ { 2 }\)

Step-by-Step Solution

Answer

The number of people in the region is 50.

Step 1: Understand the Problem

The problem requires finding the number of people in a specific region on a plane. The population density function given is \(f(x, y) = 100(y + 1)\), and the region is bounded by the curves \(x = y^2\) and \(x = 2y - y^2\). We must integrate this density function over the given region.

Step 2: Determine the Points of Intersection

To find the region bounded by the two curves, we first need to find their points of intersection. Set \(y^2 = 2y - y^2\). Solving this equation, \(y^2 + y^2 - 2y = 0\) yields \(2y^2 - 2y = 0\). Factor to get \(2y(y-1) = 0\). Thus, \(y = 0\) and \(y = 1\) are the points of intersection.

Step 3: Determine the Integral Setup

Since we have the intersection points \(y = 0\) and \(y = 1\), the bounds for \(y\) are from 0 to 1. For each \(y\) in this range, \(x\) runs from \(x = y^2\) to \(x = 2y - y^2\). Set up the double integral for the population: \[\int_{0}^{1} \int_{y^2}^{2y - y^2} 100(y + 1) \, dx \, dy.\]

Step 4: Integrate with Respect to x

Integrate the inner integral with respect to \(x\):\[\int_{y^2}^{2y - y^2} 100(y + 1) \, dx = 100(y + 1)[x]_{y^2}^{2y - y^2} = 100(y + 1)((2y - y^2) - y^2).\]Simplifying gives:\[100(y + 1)(2y - 2y^2).\]

Step 5: Simplify the Expression

Simplify the expression:\[100(y + 1)(2y - 2y^2) = 200(y + 1)(y - y^2).\]Expand the terms to get:\[200(y^2 - y^3 + y - y^2) = 200(y - y^3).\]

Step 6: Integrate with Respect to y

Now, integrate with respect to \(y\):\[\int_{0}^{1} 200(y - y^3) \, dy.\]This results in:\[200 \left[ \frac{y^2}{2} - \frac{y^4}{4} \right]_{0}^{1} = 200 \left( \frac{1}{2} - \frac{1}{4} \right) = 200 \left( \frac{1}{4} \right) = 50.\]

Step 7: Conclude the Total Population

The integral evaluates to 50, which means the number of people in the region bounded by the two curves is 50.

Key Concepts

Population Density

Population density is a crucial concept in understanding how people are distributed within a particular area. It is typically expressed as the number of individuals per unit area. In this problem, population density is given by the function \(f(x, y) = 100(y + 1)\).
This implies that at any point on the plane defined by the coordinates \(x\) and \(y\), the number of people per square mile is determined by \(100(y + 1)\). This means population increases linearly with the \(y\)-coordinate.

• If \(y = 0\), the population density is \(100\) people per square mile.
• If \(y = 1\), then density becomes \(200\) people per square mile.

Thus, knowing how to interpret and work with population density functions is integral for determining the size and distribution of a population within a region.

Integral Bounds

Integral bounds are the limits within which an integral is evaluated. They define the region over which you perform the integration. It's like setting the borders of the area where we're going to calculate quantities, such as population in this case.
In the exercise, the bounds are determined once we find where the curves \(x = y^2\) and \(x = 2y - y^2\) intersect.

• The \(y\)-bounds are determined from \(y = 0\) to \(y = 1\), based on the intersection points.
• The \(x\)-bounds vary with \(y\), from \(x = y^2\) to \(x = 2y - y^2\).

This allows us to set up a double integral that covers the specific region of interest, ensuring we calculate the population accurately.

Points of Intersection

Understanding where two curves intersect is fundamental for setting up the region of integration. These points of intersection are where the two boundaries meet, which helps define the integral's limits, or bounds.
In this problem, we find where \(x = y^2\) meets \(x = 2y - y^2\).
To find these points, set \(y^2 = 2y - y^2\). Solving \(2y^2 - 2y = 0\) by factoring gives\(2y(y - 1) = 0\), leading to solutions \(y = 0\) and \(y = 1\).

• Therefore, the points of intersection on the \(y\)-axis are at \(y = 0\) and \(y = 1\).
• These correspond to the integration limits for \(y\).

Recognizing these intersection points allows us to confine the integration to the specific area of interest between the curves.

Definite Integral

A definite integral is a calculation that provides the total amount of a quantity, like area or total population, over a specified interval. In this exercise, we use a double integral, which involves two levels of integration.
The double integral \(\int_{0}^{1} \int_{y^2}^{2y-y^2} 100(y + 1) \, dx \, dy\)
serves to calculate the total population within the defined region. Here, we:

• First integrate with respect to \(x\), holding \(y\) constant, to find the population density between \(x = y^2\) and \(x = 2y - y^2\).
• Then integrate the result with respect to \(y\), over the interval from \(y = 0\) to \(y = 1\), which sums the population across all \(y\) values.

This approach allows us to find the definite total, which is 50 people, for the region enclosed by the curves.