The overall goal of this task is to calculate the total population of a hamlet (Le. a small settlement), knowing how dense the population is at any distance from the castle at the center of the city. 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 p(D} 2 will)! . 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. 0:40 0 Population : Density ' Area P = p - A 40 Density ,0(D) = 20+Dz Color Range: Dark = Very Densely Populated Light - Sip-3:561}! Populated Show Village in: 40 c: 20 / Task: Answer the questions listed below. Again, be sure to label each piece of the required definite integrals to show how the population is being calculated locally (i.e. across the small rings near a distance D), and to specify what each symbol in the integral formula means. 1. What is the total population of the hamlet whose border is enclosed in a circle of radius 20km from the castle. Once you set up the integral, you may find the definite integral from MATLAB instead of computing the integral by hand. (Hints: You need to integrate. To do so, first find a way to approximate the small area of the ring near distance D.) 2. Suppose now that the hamlet has an "infinite" border, meaning that the population density applies for all positive distances D. Does the hamlet have a finite or an infinite population? Compare the result of this calculation to the force between the point mass and the infinite rod in Task One. Describe the similarities and differences between the two results, not just in the formulas, but specifically in the differential products, how they were constructed, and the resulting limiting functions. You may use MATLAB to find any relevant antiderivatives, but any other calculations must be done by hand. (Hint: Checking all of your calculations in MATLAB is always a good idea.)