Science:Math Exam Resources/Courses/MATH103/April 2015/Question 04
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Question 04 

Consider a swarm of ants distributed over a circular region. At distance from the centre of the region, the density of the ant population is observed to be , measured in units of one thousand per square meter. What is the total number of ants within a radius of 2m? (Work must be shown for full marks.) 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Find the integral expression for the population of ants within a radius of using its density function . 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. The most difficult part of this problem is setting up the integral. Consider a region formed between a circle of radius and a circle of radius where is infinitesimally thin. The population of ants inside this small region is given by
We can get the total population by adding up the contributions of an infinite number of these small regions in our circle of radius , or in other words, we can get the total population by integrating the above equation from to .
Use substitution:
Conclusion: There are ants 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. Let be the population of ants within a radius of with units of one thousand. Then, it has a relation with the density as follows: where represents an open ball centered at the origin with the radius and is an infinitesimal area. Writing the integral in the polar coordinate, we have
Plugging given into the integral and using the change of variable , we have and
. 