Science:Math Exam Resources/Courses/MATH200/April 2012/Question 07
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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 where A(D) is the area of the plane region D. Let D be the unit disk 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, ? 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.

Solution 

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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. The centre of D is the origin and so (a,b) = (0,0). To evaluate the integral, we use polar coordinates: where we have used the fact that (i.e. the Jacobian of the transformation from Cartesian to Polar coordinates) and . Evaluating the integral gives: Hence, the average distance from any point within D and the centre of D (the origin) is 