prepare and submit a document with your responses to this activity. Your solution should not only be an answer to the question but should include a convincing argument for the correctness of your proposed solution. This argument should contain all elements of mathematical and scientific discourse discussed on the Mathematical Discourse page.
Any claims you make should be backed up by evidence, and you should include a justification that links the evidence to the claim and incorporates the appropriate features of a definite integral.
Your evidence should include a description of the information gained from the applet from which you determined your answer. For this conceptual assessment, evidence should include the features of the applet from which you chose to construct the differential product in the integral and the bounds of the constructed definite integral. You must also verify each definite integral calculation in MATLAB.
Your justification should specifically refer to the definite integral as the accumulation of small amounts of population. This should also include a written definite integral that is labeled in detail, where each symbol of the definite integral is referred to and directly references the contextual meaning.
This activity spans two module weeks. You will submit a first draft of your document at the end of Module 4. You will then submit a final revised draft at the end of Module 5 based on feedback provided by your instructor.
The overall goal of the assessment is to calculate the total population of a hamlet (i.e. a small settlement), knowing how dense the population is at any distance from the castle at the center of the city. Then you are to then come up with a population density that redistributes a population within certain regions of the hamlet.
You are to use the following random number generator to gain your personal values “b” and “c”, to be used in the problem itself.
b= 33.93
c=13.71
f a population is uniformly distributed within an area of land, then the population has a constant density ρ. This means that the total population within the land region of size A is given by P=ρ⋅A.
Suppose that at a distance D from the hamlet’s castle (at the center of the hamlet), the density of a population is given by ρ(D)=bc+D2. This is depicted in the following GeoGebra application, where density is communicated by color: The darker colors correspond to more dense regions of the hamlet.
You are required to write an integral that calculates the total population of the hamlet distributed in this way.
Complete the following tasks:

1. Compute the total population of the hamlet whose borders are enclosed in a circle of radius b from the castle. Be sure to fully write out the integral that computes the hamlet population, to label the contextual meaning for each symbol in the integral, and to justify why the integral you wrote does compute the hamlet population. Use the Fundamental Theorem of Calculus to compute your integral, and then also verify your integral calculation in MATLAB.
2. Your b-value gives the full radius of the hamlet’s borders, meaning that the entire hamlet is contained in the circle of radius b. Determine how far out from the castle is 60% of the population. Use the density function to describe why it does or does not make sense that 60% of the population would reside within this region of the hamlet. Verify your integral calculation in MATLAB (Hint: MATLAB can be a helpful starting point for this problem).
3. Rewrite the density function so that 60% of the population is located at least b2 miles from the castle. Verify your integral calculation in MATLAB (Hint: MATLAB can be a helpful starting point for this problem).
4. Suppose that a second hamlet has the original population density ρ, but its borders extend across an infinite distance from the castle. Keep in mind that the value b no longer represents the distance from the castle to the hamlet border. Will the population of this far-reaching hamlet be finite or infinite?