Science:Math Exam Resources/Courses/MATH200/April 2012/Question 07

MATH200 April 2012

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Question 07
The average distance of a point in the plane region D to a point (a,b) is defined by ${\frac {1}{A(D)}}\iint \limits _{D}{\sqrt {(x-a)^{2}+(y-b)^{2}}}{\textrm {d}}x{\textrm {d}}y$ where A(D) is the area of the plane region D. Let D be the unit disk $\displaystyle {}1\geq {}x^{2}+y^{2}.$ Find the average distance of a point in D to the centre of D.

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Without doing any calculations, where is the centre of the unit disc, $\{(x,y)\,|\,x^{2}+y^{2}\leq 1\}$? What is the discs' area?

Hint 2
Try making a change of coordinates to simplify the evaluation of the integrals that will show up. Don't forget the Jacobian.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Since D is the unit disc, we know that it is a circle of radius 1 and so it has area π, i.e. $\displaystyle A(D)=\pi$ The centre of D is the origin and so (a,b) = (0,0). To evaluate the integral, $\iint _{D}{\sqrt {(x-0)^{2}+(y-0)^{2}}}\,dx\,dy=\iint _{D}{\sqrt {x^{2}+y^{2}}}\,dx\,dy$ we use polar coordinates: $\iint _{D}{\sqrt {x^{2}+y^{2}}}\,dx\,dy=\int _{0}^{2\pi }\int _{0}^{1}r^{2}\,dr\,d\theta$ where we have used the fact that $dx\,dy=r\,dr\,d\theta$ (i.e. the Jacobian of the transformation from Cartesian to Polar coordinates) and ${\sqrt {x^{2}+y^{2}}}=r$. Evaluating the integral gives: $\int _{0}^{2\pi }\int _{0}^{1}r^{2}\,dr\,d\theta =2\pi \int _{0}^{1}r^{2}\,dr={\frac {2\pi }{3}}.$ Hence, the average distance from any point within D and the centre of D (the origin) is ${\color {blue}{\frac {1}{A(D)}}\iint _{D}{\sqrt {x^{2}+y^{2}}}\,dx\,dy={\frac {2}{3}}.}$

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