Problem 26 - University Calculus: Early Transcendentals

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 population in the region is 50 people.

Step 1: Understand the Region

The region is bounded by the curves \( x = y^2 \) and \( x = 2y - y^2 \). Our goal is to find out where these two curves intersect, as the bounded area will be within these intersection points.

Step 2: Find Points of Intersection

To find intersections, we set the equations equal: \( y^2 = 2y - y^2 \). This results in: \( 2y^2 - 2y = 0 \) or simplified as \( y(y-1) = 0 \). Thus, the curves intersect at \( y = 0 \) and \( y = 1 \).

Step 3: Set Up the Integral

The curves intersect at \( y = 0 \) and \( y = 1 \). The function \( x = 2y - y^2 \) is greater than \( x = y^2 \) between these points. Hence, the integral for population is: \[ \int_0^1 100(y+1) \left((2y - y^2) - y^2 \right) \, dy \].

Step 4: Simplify the Expression

The expression inside the integral becomes: \((2y - y^2) - y^2 = 2y - 2y^2\). Thus, the integral is:\[ \int_0^1 100(y+1)(2y - 2y^2) \, dy \].

Step 5: Expand the Integrand

Expand \(100(y+1)(2y - 2y^2)\) to get:\[ \int_0^1 100(2y^2 + 2y) - 100(2y^3 + 2y^2) \, dy \]. Simplify to:\[ \int_0^1 100(2y^2 + 2y - 2y^3 - 2y^2)\, dy = \int_0^1 200y - 200y^3 \, dy \].

Step 6: Compute the Integral

Evaluate the integral:\[ \int_0^1 (200y - 200y^3) \, dy.\]This results in:\[ \left[ 100y^2 - 50y^4 \right]_0^1 \].Substitute the limits to compute it to be 100 - 50 = 50.

Step 7: Conclusion

The total population in the region is the result of evaluating the integral, which is 50.

Key Concepts

Integration

Integration is a fundamental concept in calculus, often used to find areas under curves or the cumulative quantity represented by a continuously varying function. In the context of this exercise, integration helps us determine the total number of people within a specified region. Here, population density is given by the function \( f(x, y) = 100(y+1) \). To calculate the total population, we integrate this density function over the area bounded by the two curves.

The integral is set up with respect to \( y \), ranging from the lower intersection point to the upper one, found in the solution as \( y = 0 \) and \( y = 1 \). The integral takes the form:

• \[ \int_0^1 100(y+1)(2y - 2y^2) \, dy \]

Integration here accounts for the continuous variation of population density across the area, multiplying the density by the differential area element to sum up small contributions over the region.

Understanding this integration process is essential as it allows for converting information about a varying density into a single, specific population count.

Curve Intersection

The curve intersection is a crucial step in analyzing problems involving region bounded by curves. It entails finding where two or more curves meet, illuminating the boundaries of the region we'll focus on. In this exercise, we seek where the curves given by the equations \( x = y^2 \) and \( x = 2y - y^2 \) intersect.

To find these points of intersection:

• Set the equations equal: \( y^2 = 2y - y^2 \)
• Rearrange to form \( 2y^2 - 2y = 0 \)
• Factor the expression to obtain solutions: \( y(y-1) = 0 \)

This results in intersection at points \( y = 0 \) and \( y = 1 \). These are the limits of integration and define the region concerning population density in this problem.

Understanding curve intersections can open up clarity in bounded region analysis, serving as a foundational step to set proper limits when calculating areas or volumes.

Bounded Region Analysis

Analyzing regions bounded by curves is vital when dealing with real-world problems involving varying characteristics over a specific area, like population density. Once intersections have been identified, this helps outline the region of interest where further calculations take place.

In this problem, the bounded region is defined between two curves \( x = y^2 \) and \( x = 2y - y^2 \). Within the intersection limits found at \( y = 0 \) and \( y = 1 \), the curve \( x = 2y - y^2 \) lies above \( x = y^2 \). This guides the formulation of the integrand for population counting.

The bounded region in focus is critical because it determines where we will apply our mathematical operations, ensuring that our calculations are physically and contextually relevant. Such analysis facilitates meaningful integration, confining our focus to precisely where it's required, resulting in accurate and efficient problem-solving.